Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics

Abstract

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., ${\Delta }^{2}x=0$. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book.

1. Introduction

The study of biharmonic submanifolds in Euclidean spaces was introduced by the author in the middle of the 1980s in his program of understanding submanifolds of finite type. Independently, G.-Y. Jiang [1] studied biharmonic maps between Riemannian manifolds as the critical points of bi-energy functionals.

If $\Delta$ is the Laplacian of a Riemannian manifold (M) isometrically immersed in a Euclidean m space (${\mathbb{E}}^m$), then M is called a biharmonic submanifold if its position vector field (x) satisfies the biharmonic equation (see [2]):

$$\begin{array}{c}\hfill {\Delta }^{2}x=0,(\mathrm{equivalently},\Delta \overrightarrow{H}=0),\end{array}$$

(1)

where $\overrightarrow{H}$ denotes the mean curvature vector of M in ${\mathbb{E}}^m$. Obviously, minimal submanifolds in ${\mathbb{E}}^m$ are automatically biharmonic submanifolds.

The author and Jiang independently proved that there are no biharmonic surfaces in the Euclidean 3-space (${\mathbb{E}}^3$), except the minimal ones. Later, in 1989, I. Dimitrić [3,4] proved in 1989 that biharmonic curves in Euclidean spaces (${\mathbb{E}}^m$) are open portions of straight lines; biharmonic finite-type submanifolds of ${\mathbb{E}}^m$ are minima, biharmonic hypersurfaces of ${\mathbb{E}}^m$ with, at most, two distinct principal curvatures are minimal, and pseudo-umbilical submanifolds of ${\mathbb{E}}^m$ with $dimM\ne 4$ are minimal. In addition, the author showed that biharmonic submanifolds in ${\mathbb{E}}^m$ are minimal whenever they lie in a hypersphere of ${\mathbb{E}}^m$ (see, e.g., [2] page 181 or [5] Corollary 7.2).

In 1991, the author proposed the following conjecture (see, e.g., [2,6,7]):

Chen’s Biharmonic Conjecture. Biharmonic submanifolds of Euclidean spaces are minimal, which is well known today as Chen’s (biharmonic) conjecture. Chen’s conjecture was proven to be true by T. Hasanis and T. Vlachos for any hypersurface in ${\mathbb{E}}^4$ [8] (see also [9]). Later, Y. Fu [10,11] proved that Chen’s conjecture holds for hypersurfaces with three distinct principal curvatures for arbitrary m. In addition, S. Montaldo, C. Oniciuc, and A. Ratto [12] proved Chen’s conjecture to be true for G-invariant hypersurfaces. In contrast to Euclidean submanifolds, Chen’s biharmonic conjecture is not necessarily true for submanifolds in pseudo-Euclidean spaces, which was achieved by the author and Ishikawa in [13,14]. In fact, the author and Ishikawa have constructed many examples of non-minimal, proper biharmonic surfaces in pseudo-Euclidean spaces (${\mathbb{E}}_s^4$) with an index of $s=1,2$, or 3. On the other hand, for surfaces in a Minkowski 3-space, the author and Ishikawa [13,14] proved that biharmonic surfaces are minimal. In addition, Arvanitoyeorgos et al. [15] showed that biharmonic Lorentian hypersurfaces in a Minkowski 4-space are minimal.

The study of biharmonic maps can be regarded as a special case of a program in understanding the geometry of k-polyharmonic maps, which was originally proposed by J. Eells and L. Lemaire [16] in 1983. Biharmonic maps are a generalization of biharmonic functions that have important applications in elasticity and hydrodynamics, among others. Biharmonic maps are also a generalization of harmonic maps, which include important objects such as minimal submanifolds, geodesics, harmonic functions, and Riemannian submersions with minimal fibers.

From the viewpoint of k-harmonic maps, it is natural to consider a biharmonic map ($\phi$) as a critical point of the bi-energy functional. G.-Y. Jiang [17] proved that a smooth map ($\phi$) is biharmonic if and only if its bitension field (${\tau }_2\left(\phi \right)$) vanishes identically, i.e., ${\tau }_2\left(\phi \right)=0$. In 2002, R. Caddeo and S. Montaldo [18] pointed out that, for a Euclidean submanifold, ”${\tau }_2\left(\phi \right)=0$ holds identially” is equivalent to “$\Delta \overrightarrow{H}=0$ holds identially”. Therefore, for Euclidean submanifolds, both definitions of biharmonicity of the author and G.-Y. Jiang coincide.

Since the beginning of this century, there has also been a lot of research work done on biharmonic submanifolds in Riemannian manifolds other than in Euclidean spaces (see, e.g., [19,20,21,22,23,24]). A comprehensive survey of important results on Chen’s biharmonic conjecture and many related topics was presented by Y.-L. Ou and the author in their 2020 book.

Thus, the main purpose of this review paper is to provide a detailed survey of recent advances in Chen’s conjecture and generalized Chen biharmonic conjectures made after the publication of Ou and Chen’s book. Many recent results in several related subjects are also presented in this paper. The author intends this review paper to provide a useful reference for doctoral students and researchers who are interested in the theory of biharmonic and biconservative submanifolds.

2. Basics on Submanifolds

Let M be a differential manifold and E be a vector bundle over M. $\mathfrak{X}\left(E\right)$ denotes the space of all (smooth) vector fields in E. In particular, we denote the space of all smooth vector fields tangent to M as $\mathfrak{X}\left(TM\right)$.

Let $\varphi :M\to {\mathbb{E}}^m$ denote an isometric immersion of a Riemannian manifold (M) into ${\mathbb{E}}^m$. Let ∇ and $\tilde{\nabla }$ denote the Levi–Civita connections of M and ${\mathbb{E}}^m$, respectively. Then, we have the Gaussian and Weingarten formulas given respectively by (cf. [5,25])

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_X\xi =-A_{\xi }X+D_X\xi ,\hfill \end{array}$$

(3)

where X and Y are vector fields tangent to M and $\xi$ is a vector field normal to M. We call $h, A,$ and D the second fundamental form, the shape operator, and the normal connection of M, respectively. The second fundamental form (h) and the shape operator (A) are related by

$$\begin{array}{c}\hfill \langle h(X,Y),\xi \rangle =\langle A_{\xi }X,Y\rangle ,\end{array}$$

(4)

where $\langle ·,·\rangle$ denotes the inner product.

The mean curvature vector field ($\overrightarrow{H}$) is defined by

$$\begin{array}{c}\hfill \overrightarrow{H}=\left({\displaystyle \frac{1}{n}}\right)\mathrm{Trace}h.\end{array}$$

(5)

The submanifold (M) is said to have a parallel mean curvature vector if $D\overrightarrow{H}=0$ holds identically (see, e.g., [26]). M is called totally geodesic if $h=0$ holds and totally umbilical if $h=g\otimes \overrightarrow{H}$ holds. The submanifold (M) is said to be pseudo-umbilical if $A_{\overrightarrow{H}}$ is proportional to the identity map (I).

Gauss and Codazzi’s equations are given respectively by

$$\begin{array}{c}\langle R(X,Y)Z,W\rangle =\langle h(X,W),h(Y,Z)\rangle -\langle h(X,Z),h(Y,W)\rangle ,\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_{X}h\right)(Y,Z)=\left({\overline{\nabla }}_{Y}h\right)(X,Z),\end{array}$$

(7)

where $X,Y,Z$, and W are vector fields tangent to M; R denotes the Riemann curvature tensor of M; and the covariant derivative ($\overline{\nabla }h$) of h is given by

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_{X}h\right)(Y,Z)=D_{X}h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z).\end{array}$$

(8)

Ricci’s equation is given by

$$\begin{array}{c}\hfill \langle R^D(X,Y)\xi ,\zeta \rangle =\langle [A_{\xi },A_{\zeta }]\left(X\right),Y\rangle ,\end{array}$$

(9)

where X and Y are vector fields tangent to M, $\xi$ and $\zeta$ are vector fields normal to M, and $R^D$ is the normal curvature tensor defined by

$$\begin{array}{c}\hfill R^D(X,Y)\xi =D_X{D}_Y\xi -D_Y{D}_X\xi -D_{[X,Y]}\xi .\end{array}$$

(10)

The Laplacian ($\Delta$) of M is given by

$$\Delta f=\mathrm{div}(\nabla f)=\Delta f=\sum _{i,j=1}^n{g}^{ij}\left\{{\displaystyle \frac{{\partial }^{2}f}{\partial x^i\partial x^j}}-\sum _{k=1}^n{\mathsf{\Gamma }}_{ij}^k{\displaystyle \frac{\partial f}{\partial x^k}}\right\},$$

where $\{x_1,\dots ,x_n\}$ is a local coordinate system of M, $\nabla f$ is the gradient of f, $\mathrm{div}\left(X\right)$ is the divergence of X, and ${\mathsf{\Gamma }}_{ij}^k(i,j,k=1,\dots ,n)$ are the Christoffel symbols.

3. Submanifolds of Finite Type and of Null 2 Type

Let M be a Riemannian manifold with a Laplacian of $\Delta$. A map ($\varphi :M\to {\mathbb{E}}^m$) of M into ${\mathbb{E}}^m$ is said to be of finite type if $\varphi$ has a finite spectral resolution (cf. [27]):

$$\begin{array}{c}\hfill \varphi =c+\sum _{i=1}^k{\varphi }_{t_i},\Delta {\varphi }_{t_i}={\lambda }_{t_i}{\varphi }_{t_i},\end{array}$$

(11)

with ${\lambda }_{t_1}<{\lambda }_{t_2}<\dots <{\lambda }_{t_k}$, where c is a constant vector in ${\mathbb{E}}^m$ and ${\varphi }_{t_1},\dots ,{\varphi }_{t_k}$ are non-constant ${\mathbb{E}}^m$-valued maps. Otherwise, the map ($\varphi$) is called an infinite type. When the spectral resolution (11) contains exactly k non-constant terms, the map ($\varphi$) is of k type.

A finite-type map ($\varphi :M\to {\mathbb{E}}^m$) is called null if its spectral resolution (11) contains a non-constant harmonic component (${\varphi }_0$), i.e., a non-constant component (${\varphi }_0$) with $\Delta {\varphi }_0=0$. In particular, the map is said to be of null 2 type it the map ($\varphi$) admits the following spectral decomposition:

$$\varphi ={\varphi }_0+{\varphi }_q,$$

(12)

where ${\varphi }_0$ and ${\varphi }_q$ are non-zero vector functions satisfying $\Delta {\varphi }_0=0$ and $\Delta {\varphi }_q=\lambda {\varphi }_q$ for some non-zero real number ($\lambda$) (see [6,27,28,29,30]). It follows from (12) that the position vector (x) satisfies

$$\begin{array}{c}\hfill {\Delta }^2\varphi =\lambda \Delta \varphi .\end{array}$$

(13)

According to the theory of finite type submanifolds, the family of null 2-type submanifolds is quite interesting (see the book [27]).

The mean curvature vector field ($\overrightarrow{H}$) of $\varphi :M\to {\mathbb{E}}^m$ satisfies the following Beltrami formula (see, for instance, page 44 of [27]):

$$\begin{array}{c}\hfill \Delta \varphi =n\overrightarrow{H},\end{array}$$

(14)

which shows that $\varphi$ is biharmonic if and only if it satisfies $\Delta \overrightarrow{H}=0$ identically (or, equivalently, ${\Delta }^2\varphi =0$ identically).

The following characterization of biharmonic submanifolds in ${\mathbb{E}}^m$ is an easy consequence of the expression of $\Delta \overrightarrow{H}$ given in [5,27,29,30].

Theorem 1.

If $\varphi :M\to {\mathbb{E}}^m$ is an isometric immersion, then M is a biharmonic submanifold if and only if it satisfies

$$\begin{array}{c}{\Delta }^D\overrightarrow{H}=\sum _{i=1}^{n}h\left(A_{\overrightarrow{H}}{e}_i,e_i\right),\hfill \end{array}$$

(15)

$$\begin{array}{c}\hfill n\nabla \langle \overrightarrow{H},\overrightarrow{H}\rangle +4\mathrm{Trace}\left(A_{D\overrightarrow{H}}\right)=0,\end{array}$$

(16)

where $\{e_1,\dots ,e_n\}$ is a local orthonormal tangent frame on M.

From Condition (15), it follows that each biharmonic submanifold of ${\mathbb{E}}^m$ is minimal if its mean curvature vector field is parallel.

4. Biharmonic Map and Bitension

Let $\varphi :(M,g)\to (N,\tilde{g})$ be a smooth map between two Riemannian manifolds. $\varphi$ is called a harmonic map [31] if it is a critical point of the following energy functional:

$$\begin{array}{c}\hfill E\left(\varphi \right)={\displaystyle \frac{1}{2}}{\int }_M{\left|\mathrm{d}\varphi \right|}^2\mathrm{d}{v}_g,\end{array}$$

(17)

It is characterized by the vanishing of the tension field:

$$\tau \left(\varphi \right)=\mathrm{Trace}(\nabla \mathrm{d}\varphi )=0.$$

On the other hand, in 1983, Eells and Lemaire [16] proposed a problem considering k-harmonic maps, which are critical maps of the following functional for a smooth map ($\varphi :(M,g)\to (N,\tilde{g})$):

$$E_k\left(\varphi \right)={\int }_M|e_k\left(\varphi \right)|\mathrm{d}{v}_g),k=1,2,\dots ,$$

(18)

where $e_k\left(\varphi \right)$ is defined by

$$e_k\left(\varphi \right)={\displaystyle \frac{1}{2}}{\parallel {(d+d^{*})}^k\varphi \parallel }^2.$$

A biharmonic map is a smooth map ($\varphi :(M,g)\to (N,\tilde{g})$) that is a critical point of the bi-energy functional:

$$E_2\left(\varphi \right)={\displaystyle \frac{1}{2}}{\int }_M{\left|\tau \left(\varphi \right)\right|}^2\mathrm{d}{v}_g.$$

(19)

G.-Y. Jiang [17] derived the first variational formula of a bi-energy functional and obtained the following biharmonic map equation as the Euler–Lagrange equation for bi-energy:

$${\tau }_2\left(\varphi \right)={\Delta }^{\varphi }\tau \left(\varphi \right)-\mathrm{Trace}\left({\mathrm{R}}^N(\mathrm{d}\varphi (·),\tau \left(\varphi \right))\mathrm{d}\varphi (·)\right)=0,$$

(20)

where ${\Delta }^{\varphi }Z=\mathrm{Tr}({\nabla }^{\varphi }{\nabla }^{\varphi }-{\nabla }_{{\nabla }^M}^{\varphi })Z$.

The biharmonic equation (20) implies the following (cf., e.g., [22] or [5], page 153).

Proposition 1.

If $\varphi :M^n\to (N^m,\tilde{g})$ is an isometric immersion, then ϕ is a biharmonic submanifold if and only if we have the following:

$$\begin{array}{c}n\nabla \langle \overrightarrow{H},\overrightarrow{H}\rangle +4\mathrm{Trace}(\nabla A_{D\overrightarrow{H}})=-4\sum _{i=1}^n{\left(R^N(e_i,\overrightarrow{H})e_i\right)}^T,\hfill \end{array}$$

(21)

$$\begin{array}{c}\hfill {\Delta }^D\overrightarrow{H}-\sum _{i=1}^{n}h(A_{\overrightarrow{H}}{e}_i,e_i)=\sum _{i=1}^n{\left(R^N(e_i,\overrightarrow{H})e_i\right)}^{\perp },\end{array}$$

(22)

where $R^N$ the Riemann curvature tensor of $(N^m,\tilde{g})$ and $\{e_1,\dots ,e_m\}$ is an orthonormal local frame of $M^n$.

Note that Equations (21) and (22) reduce to Equations (15) and (16), respectively, when the ambient space is a Euclidean m space.

A biharmonic submanifold of a Riemannian manifold is a submanifold whose inclusion map is biharmonic. A biharmonic submanifold is called proper if it is not minimal.

An interesting problem related to Chen’s biharmonic conjecture was proposed in 2001 by Caddeo, Montaldo, and Oniciuc in [32], described as follows.

• Generalized Chen Biharmonic Conjecture: Every biharmonic submanifold in a Riemannian manifold of non-positive sectional curvature is minimal.

In 2012, Y.-L. Ou and L. Tang [33] showed that this generalized Chen’s biharmonic conjecture is false for all codimensions. After that, generalized Chen biharmonic conjecture was modified by Caddeo, Montaldo, and Oniciuc as follows.

• (Modified) Generalized Chen Biharmonic Conjecture: Any biharmonic submanifold of a Riemannian manifold of non-positive constant sectional curvature is always minimal.

5. Recent Advances in Chen’s Biharmonic Conjecture

Today, Chen’s biharmonic conjecture is still widely open, although several partial results have been proven for low dimensions some additional geometric conditions (e.g., [34,35,36,37]). Since the most fundamental biharmonic submanifolds are harmonic hypersurfaces of Euclidean spaces, many mathematicians have focused on such cases during the last four decades. Thus, we discuss biharmonic hypersurfaces in Euclidean spaces in this section.

5.1. Biharmonic Hypersurfaces in ${\mathbb{E}}^6$

Assume that $M^n$ is a biharmonic hypersurface in ${\mathbb{E}}^{n+1}$. Recall that the author and Jiang [1] proved independently that any biharmonic surface ($M^2$) in ${\mathbb{E}}^3$ is minimal. Applying the author’s method, I. Dimitrić extended this result to hypersurfaces in ${\mathbb{E}}^{n+1}$ with, at most, two distinct principal curvatures (see Dimitrić’s doctoral thesis [3] at Michigan State University). In his thesis, Dimitrić also showed that any biharmonic submanifold of finite type in a Euclidean space is minimal. In 1995, T. Hasanis and T. Vlachos [8] made significant progress and settled Chen’s conjecture for the case of $n=3$. In 1998, F. Defever [9] reproved the conjecture for $n=3$ using a new tensorial analysis approach. In 2021, Y. Fu, M.-C. Hong, and X. Zhan [38] took an important step by settling the conjecture for hypersurfaces in ${\mathbb{E}}^5$—namely, they proved the following.

Theorem 2

([38]). Any biharmonic hypersurface in ${\mathbb{E}}^5$ is minimal.

Recently, Theorem 2 was improved by Y. Fu, M.-C. Hong, and X. Zhang as follows.

Theorem 3

([39]). Any biharmonic hypersurface in ${\mathbb{E}}^6$ is minimal.

Furthermore, Y. Fu proved the following result in 2015.

Theorem 4

([11]). Every biharmonic hypersurface in ${\mathbb{E}}^{n+1}(n\ge 5)$ with, at most, three distinct principal curvatures is minimal.

5.2. Properly Immersed and Weakly Convex Biharmonic Submanifolds

Properly immersed submanifolds are defined as follows.

Definition 1.

An immersed submanifold (M) of a Riemannian manifold is called properly immersed if the immersion is a proper map, i.e., the preimage of each compact set in the ambient space is compact in M.

Note that the properness (of the immersion) implies the completeness of the immersed submanifold.

The next theorem of K. Akutagawa and S. Maeta also supports Chen’s biharmonic conjecture.

Theorem 5

([34]). Any properly immersed biharmonic submanifold in ${\mathbb{E}}^m$ is minimal.

Definition 2.

A hypersurface of a Riemannian manifold is called weakly convex if it has non-negative principal curvatures.

Theorem 6

([35]). Any weakly convex biharmonic hypersurface of ${\mathbb{E}}^{n+1}$ is minimal.

5.3. Biharmonic Hypersurface with a Recurrent Ricci Operator in ${\mathbb{E}}^{n+1}$

If T is a tensor on a Riemannian manifold (M), then T is called recurrent if there is a certain 1-form ($\eta$) on M such that ${\nabla }_{X}T=\eta \left(X\right)T$ holds for any vector (X) tangent to M.

E. Abedi and N. Mosadegh proved the following partial solution to Chen’s biharmonic conjecture.

Theorem 7

([40]). If $M^n$ is a biharmonic hypersurface in ${\mathbb{E}}^{n+1}$ with a recurrent Ricci operator, then $M^n$ is minimal.

5.4. Holonomic Biharmonic Hypersurfaces in Euclidean Spaces

A hypersurface (M) of ${\mathbb{E}}^{n+1}$ is called holonomic if the principal curvature lines on M are defined on an open subdomain in a coordinate system—in other words, if the principal curvature directions define an integrable net (see, e.g., [41]).

Very recently, H. Bibi, M Soret, and M. Ville [42] studied biharmonic hypersurfaces of a Euclidean space that are holonomic. They proved the following.

Theorem 8.

Holonomic biharmonic hypersurfaces of ${\mathbb{E}}^{n+1}$ are minimal.

5.5. Stability and the Index of Biharmonic Hypersurfaces in Euclidean Spaces

In [43], Y.-L. Ou established an explicit second variation formula for biharmonic hypersurfaces in a Riemannian manifold, which is similar to that of minimal hypersurfaces. By applying the second variation formula, he computed the normal stability index of the known biharmonic hypersurfaces in a Euclidean sphere. To prove the nonexistence of unstable proper biharmonic hypersurface in Euclidean space, he added another special case to support Chen’s biharmonic conjecture.

6. PNMCV Biharmonic Submanifolds in Euclidean Spaces

At the beginning of the 1980s, the author initiated the study of submanifolds with a parallel normalized mean curvature vector. Recall that a submanifold of a Riemannian manifold is said to have a parallel normalized mean curvature vector field (i.e., PNMCV) if the mean curvature vector ($\overrightarrow{H}$) is non-zero and the unit normal vector field ($\xi =\overrightarrow{H}/|\overrightarrow{H}|$) is parallel in the normal bundle ($T^{\perp }M$) (see [44]). Clearly, any non-minimal hypersurface has a parallel normalized mean curvature vector. Furthermore, the family of submanifolds with a parallel normalized mean curvature vector contains the family of submanifolds with a parallel mean curvature vector ($\overrightarrow{H}\ne 0$) as a proper subset. This is simply because a submanifold has a parallel mean curvature vector ($\overrightarrow{H}\ne 0$) if and only if it has a parallel normalized mean curvature vector with constant mean curvature.

Next, we provide another solution to Chen’s biharmonic conjecture from [45].

Theorem 9.

There are no biharmonic surfaces in ${\mathbb{E}}^m$ with a parallel normalized mean curvature vector.

Remark 1.

When $n=4$, Theorem 9 is due to [46].

In [47], R. Y. Şen and N. C. Turgay proved the following.

Theorem 10.

There are no biharmonic submanifolds in ${\mathbb{E}}^5$ with a parallel normalized mean curvature vector.

7. $\delta \mathbf{\left(}\mathit{r}\mathbf{\right)}$-Ideal Biharmonic Hypersurfaces in ${\mathbb{E}}^{\mathit{m}}$

The theory of $\delta$ invariants was invented by the author in the early 1990s (see [25,48,49]). In particular, he introduced $\delta \left(k\right)$-ideal submanifolds.

Let us recall the definition of $\delta$ invariants of Riemannian manifolds from [25,49]. $K\left(\pi \right)$ denotes the sectional curvature of a plane section ($\pi \subset T_{p}M(p\in M)$) of a Riemannian n manifold (M). For a given orthonormal basis ($e_1,\dots ,e_n$) of the tangent space ($T_{p}M$) at p, the scalar curvature ($\tau$) at p is given by

$$\begin{array}{c}\hfill \tau \left(p\right)=\sum _{i<j}K(e_i\wedge e_j).\end{array}$$

(23)

If $L_r$ is a linear r subspace of $T_{p}M$ with $r\ge 2$ and if $\{e_1,\dots ,e_r\}$ is an orthonormal basis of $L_r$, then the scalar curvature ($\tau \left(L_r\right)$) of $L_r$ is given by

$$\begin{array}{c}\hfill \tau \left(L_r\right)=\sum _{\le \alpha <\beta \le r}K(e_{\alpha }\wedge e_{\beta }).\end{array}$$

(24)

For $r\in [2,n-1]$, we define the δ-invariant $\delta \left(r\right)$ as

$$\begin{array}{c}\hfill \delta \left(r\right)\left(p\right)=\tau \left(p\right)-inf\left\{\tau \left(L_r\right)\right\},\end{array}$$

(25)

where $L_r$ runs over all linear r subspaces of $T_{p}M$.

For any n-dimensional submanifold ($M^n$) in ${\mathbb{E}}^m$, the author proved the following general optimal inequality (see [25,49]):

$$\begin{array}{c}\hfill \delta \left(r\right)\le {\displaystyle \frac{n^2(n-r)}{2(n-r+1)}}{\parallel \overrightarrow{H}\parallel }^2.\end{array}$$

(26)

Submanifolds ($M^n$) in ${\mathbb{E}}^m$ are called $\delta \left(r\right)$-ideal if they satisfy the equality case of (26) identically. In physical terms, an ideal submanifold of ${\mathbb{E}}^m$ is nothing but a submanifold that receives the least possible tension from its ambient space. Ideal submanifolds have many interesting properties, and they have been studied by many mathematicians during the last three decades (see [25] for details).

The author and M. I. Munteanu [50] proved the following result.

Theorem 11.

Any $\delta \left(2\right)$-ideal or $\delta \left(3\right)$-ideal biharmonic hypersurface in ${\mathbb{E}}^{n+1}$ is minimal for any $n\ge 3$ or $n\ge 4$ respectively.

In [51], Deepika and A. Arvanitoyeorgos proved the following.

Theorem 12.

Any $\delta \left(r\right)$-ideal biharmonic hypersurface in ${\mathbb{E}}^{n+1}$$(n\ge 3)$ with, at most, $r+1$ distinct principal curvatures is minimal.

8. $\lambda$-Harmonic Hypersurfaces in Euclidean Space

8.1. $(k,\ell ,\lambda )$-Harmonic Maps, $\lambda$-Harmonic Submanifolds, and Null 2-Type Submanifolds

The author extended k-harmonic maps to $(k,\ell ,\lambda )$-harmonic maps as follows (see [27], page 277 for more details).

Definition 3.

Given two integers (k and ℓ with $k>\ell \ge 1$) and a real number (λ), we call a map, i.e.,

$$\varphi :(M,g)\to (\tilde{M},\tilde{g})$$

between Riemannian manifolds $(k,\ell ,\lambda )$-harmonic if it is a critical point of the $(k,\ell ,\lambda )$-energy functional ($E_{k,\ell ,\lambda }\left(\varphi \right)$) given by

$$E_{k,\ell ,\lambda }\left(\varphi \right)=E_k\left(\varphi \right)+\lambda E_{\ell }\left(\varphi \right)$$

(27)

for variations (${\varphi }_t:(-ϵ,ϵ)\times M\to \tilde{M}$) that satisfy ${\varphi }_0=\varphi$ and $E_1\left(\varphi \right)=E\left(\varphi \right)$.

If we restrict ${\varphi }_t$ to normal variations only, then the critical points of the $(k,\ell ,\lambda )$-energy functional are called$\left(k,\ell ,\lambda \right)$-minimal.

Definition 4.

We call a $(2,1,\lambda )$-harmonic map a λ-biharmonic map, and we call a λ-harmonic map $\varphi :(M,g)\to (\tilde{M},\tilde{g})$ a $\lambda$-harmonic submanifold if ϕ is an isometric immersion.

We know from Definition 3 and (18) that a map ($\varphi :(M,g)\to {\mathbb{E}}^m$) is $(k,\ell ,\lambda )$-harmonic if and only if we have the following:

$$\begin{array}{c}\hfill {(-1)}^{k-1}{\Delta }^k\varphi +{(-1)}^{\ell -1}\lambda {\Delta }^{\ell }\varphi =0.\end{array}$$

(28)

Clearly, it follows from Definition 3 and (28) that a map ($\varphi :(M^n,g)\to {\mathbb{E}}^m$) is $\lambda$-harmonic if and only if the following condition is satisfied:

$$\begin{array}{c}\hfill {\Delta }^2\varphi =\lambda \Delta \varphi .\end{array}$$

(29)

Consequently, after combining (29) with Bletrami’s formula (see [27,30,52])

$$\Delta \varphi =-n\overrightarrow{H},$$

and using the definition of $\lambda$-harmonic submanifolds, we may conclude the following:

“Important Fact: λ-harmonic submanifolds of ${\mathbb{E}}^m$ are exactly submanifolds of null 2 type”.

Remark 2.

For further results on λ-harmonic maps and λ-harmonic submanifolds, we refer readers to [27].

8.2. $\lambda$-Harmonic Submanifolds in Euclidean Spaces

Recently, D. Yang and X. Zhan [53] studied $\lambda$-biharmonic hypersurfaces in ${\mathbb{E}}^6$ with $\lambda \le 0$ or $\lambda >0$. They proved the next two results.

Theorem 13

([53]). Every λ-biharmonic hypersurface in ${\mathbb{E}}^6$ with $\lambda \le 0$ is minimal.

Theorem 14

([53]). Every λ-biharmonic hypersurface in ${\mathbb{E}}^6$ with $\lambda >0$ is either minimal or it has a non-zero constant mean curvature and constant scalar curvature.

Definition 5.

A hypersurface in a spatial form is said to be linear Weingarten if its scalar curvature (τ) and mean curvature ($\alpha =|\overrightarrow{H}|$) satisfy $a\tau +b\alpha =c$ for some constants ($a,b,c$).

In [54], D. Yang, J. Zhang, and Y. Fu proved the following results.

Theorem 15

([54]). Every linear Weingarten λ-biharmonic hypersurface of ${\mathbb{E}}^{n+1}$ with $n<7$ is minimal.

Theorem 15 implies the following.

Corollary 1

([54]). Every linear Weingarten biharmonic hypersurface of ${\mathbb{E}}^{n+1}$ with $n<7$ is minimal.

9. Biharmonic Hypersurfaces in Pseudo-Euclidean Spaces ${\mathbb{E}}_{\mathit{s}}^{\mathit{m}}$

For biharmonic surfaces (${\mathbb{E}}_s^3$$(s=1,2)$), the author and Ishikawa [13] proved the following.

Theorem 16.

For $r=0,1,2$, every biharmonic pseudo-Riemannian surface (M) in ${\mathbb{E}}_r^3$ is always minimal.

The next two theorems, obtained by the author and Ishikawa, show that Chen’s biharmonic conjecture cannot be true for biharmonic submanifolds in pseudo-Euclidean spaces with an index of $s\ge 1$.

For biharmonic surfaces (${\mathbb{E}}_s^4$$(s=0,1,2)$), the author and Ishikawa [14] proved the following.

Theorem 17

([13]). A marginally trapped surface in ${\mathbb{E}}_1^4$ is biharmonic if and only if it is congruent to a flat surface with a flat normal connection defined by

$$\begin{array}{c}\hfill \varphi (x,y)=\left(\phi \right(x,y),u(x,y),v(x,y),\phi (x,y\left)\right),\end{array}$$

(30)

where $\phi (x,y)$ is a smooth function satisfying $\Delta \phi \ne 0$ and ${\Delta }^2\phi =0$ and $u(x,y)$ is the real part of a holomorphic function in $z=x+\mathrm{i}y$ that satisfies the Eikonal equation: $|\nabla u|=1$.

Theorem 18

([14]). If $\varphi :M\to {\mathbb{E}}_r^4(r=0,1,2)$ is a space-like surface in ${\mathbb{E}}_s^4$, then M is a biharmonic surface with non-zero constant mean curvature if and only if one of the following two types of flat surfaces with flat normal connection:

(1)

M is congruent to a surface in ${\mathbb{E}}_1^4$ defined by

$$\begin{array}{c}\hfill \varphi (u,v)={\displaystyle \frac{1}{6}}\left(a^2{u}^3,3a{u}^2,6u-a^2{u}^3,6v\right);\end{array}$$

(31)

(2)

M is congruent to a surface in ${\mathbb{E}}_2^4$ defined by

$$\begin{array}{c}\hfill \varphi (u,v)={\displaystyle \frac{1}{6}}\left(a^2{u}^3,3a{u}^2,6u+a^2{u}^3,6v\right),\end{array}$$

(32)

where a is a non-zero real number.

Next, we present the following definitions.

Definition 6

([55]). If M is a pseudo-Riemannian submanifold of a pseudo-Riemannian manifold (N), then M is called quasi-minimal if it has a non-zero mean curvature vector that satisfies $\langle \overrightarrow{H},\overrightarrow{H}\rangle =0$ identically.

Definition 7

([56]). A space-like submanifold of a Lorentzian manifold is called marginally trapped if it is quasi-minimal.

For quasi-minimal biharmonic Lorentzian flat surfaces in ${\mathbb{E}}_2^4$, we have the following.

Theorem 19

([57]). A quasi-minimal immersion $(\varphi :M\to {\mathbb{E}}_2^4)$ of a flat Lorentz surface into ${\mathbb{E}}_2^4$ is biharmonic if and only if it is congruent to one of the following two types of surfaces:

(1)

A surface defined by

$$\begin{array}{c}\hfill \varphi (x,y)={\displaystyle \frac{1}{\sqrt{2}}}\left(\phi (x,y),x+y,x-y,\phi (x,y)\right),\end{array}$$

(33)

where $\phi (x,y)$ is a smooth function defined on an open domain $(U\subset {\mathbb{E}}_1^2)$ satisfying ${\phi }_{xy}\ne 0$ and ${\phi }_{xxyy}=0$.

(2)

A surface defined by

$$\begin{array}{c}\hfill \varphi (x,y)=z\left(x\right)y+w\left(x\right),\end{array}$$

(34)

where z is a null curve lying in the light cone $(\mathcal{LC}\subset {\mathbb{E}}_2^4)$ and w is a null curve satisfying $〈z^{\prime },w^{\prime }〉=0$ and $〈z,w^{\prime }〉=-1$.

In 2023, L. Du and X. Yuan proved the following.

Theorem 20

([58]). Let $M_r^n$ be a biharmonic hypersurface in a pseudo-Euclidean space with a diagonalizable shape operator. If either of the curvature operators, Ricci operator, Jacobi operator, or shape operator of $M_r^n$ is recurrent, then $M_r^n$ is minimal

Remark 3.

For further results in this respect, we refer to Ou and Chen’s 2020 book [5].

10. Biharmonic Submanifolds in Real Space Forms

For proper biharmonic submanifolds of spheres, Balmuş, Montaldo, and Oniciuc [20] made the following two conjectures (see also [59]).

BMO Conjecture 1.

Any proper biharmonic submanifold of a sphere has constant mean curvature.

BMO Conjecture 2.

The only proper biharmonic hypersurface of the unit $(n+1)$ sphere $S^{n+1}$ is an open portion of $S^n\left({\displaystyle \frac{1}{\sqrt{2}}}\right)$ or of $S^p\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\times S^q\left({\displaystyle \frac{1}{\sqrt{2}}}\right)$ with $p+q=n,p\ne q$.

10.1. Biharmonic Submanifolds with, at Most, Three Distinct Principal Curvatures

By a real space form, we mean a Riemannian manifold with constant sectional curvature. For biharmonic hypersurfaces in a real space form with, at most, three distinct principal curvatures, Yu Fu proved the following.

Theorem 21

([60]). If M is a proper biharmonic hypersurface in a real space form with, at most, three distinct principal curvatures, then M has a constant mean curvature.

Theorem 22

([60]). Let M be a proper biharmonic hypersurface with, at most, three distinct principal curvatures in $S^{n+1}\left(1\right)$. Then, M has a constant mean curvature.

Theorem 23

([60]). Let M be an orientable proper biharmonic closed hypersurface with, at most, three distinct principal curvatures in $S^{n+1}\left(1\right)$. Then, M is either a small hypersphere $\left(S^n\left(\frac{1}{\sqrt{2}}\right)\right)$ or a generalized Clifford torus $(S^{n_1}\left(\frac{1}{\sqrt{2}}\right)\times S^{n_2}\left(\frac{1}{\sqrt{2}}\right))$ with $n_1+n_2=n$ and $n_1\ne n_2$.

In [61], S. Andronic, Y. Fu, and C. Oniciuc presented an alternate proof of Theorem 23.

10.2. Biharmonic Hypersurfaces with Constant Scalar Curvature

In 2018, Y. Fu and M.-C. Hong proved the following.

Theorem 24

([62]). Let $M^n$ be a biharmonic hypersurface of a real space form $\left(R^{n+1}\left(c\right)\right)$ with, at most, six distinct principal curvatures. If $M^n$ has a constant scalar curvature, then $M^n$ has a constant mean curvature.

For biharmonic hypersurfaces with constant scalar curvature, S. Maeta and Y.-L. Ou proved the following.

Theorem 25

([63]). An Einstein hypersurface $\left(M^n(n\ge 3)\right)$ of a real space form $\left(R^{n+1}\left(c\right)\right)$ is biharmonic if and only if it is either minimal or ${\parallel A\parallel }^2=nc$. Moreover, in the latter case, the hypersurface has positive scalar curvature, as expressed by $\tau =n(n-2)c+n^2{|\overrightarrow{H}|}^2>0$.

Theorem 26.

A compact hypersurface ($M^n$) of $S^{n+1}$ with constant scalar curvature is biharmonic if and only if it is either minimal or it has non-zero constant mean curvature with ${\parallel A\parallel }^2=n$.

10.3. Biharmonic Submanifolds with Constant Mean Curvature

For biharmonic hypersurfaces in a real space form ($R^5\left(c\right)(c>0)$), Z. Guan, H. Li, and L. Vrancken [64] proved the following.

Theorem 27

([64]). Every biharmonic hypersurface in $R^5\left(c\right)(c\ne 0)$ has constant mean curvature.

In 2023, Y. Fu, M.-C. Hong, and X. Zhan improved Theorem 27 as follows.

Theorem 28

([39]). Any biharmonic hypersurface in a real space form $\left(R^6\left(c\right)(c\ne 0)\right)$ has constant mean curvature.

Remark 4.

When $c>0$, Theorem 28 provides a positive answer to BMO Conjecture 1 for $n=5$ and codimension one. When $c<0$, Theorem 28 implies that five-dimensional biharmonic hypersurfaces in $H^6\left(c\right)$ are minimal, solving the generalized Chen conjecture for $n=5$ and codimension one.

For an orientable hypersurface ($M^n$), one can choose an orientation for $M^n$ such that the mean curvature function ($\alpha =|\overrightarrow{H}|$) is non-negative.

X. Wang and L. Wu [65] proved the following.

Theorem 29

([65], Corollary 1.5). If $M^n(n\ge 3)$ is a CMC proper biharmonic hypersurface in $S^{n+1}$, then locally, $M^n$ is one of the following:

(1)

$\alpha =1$ and $M=S^n\left(\frac{1}{\sqrt{2}}\right)$;

(2)

$\alpha \in (0,\frac{n-2}{n}]$, and $\alpha =\frac{n-2}{n}$ holds if and only if $M^n=S^{n-1}\left(\frac{1}{\sqrt{2}}\right)\times S^1\left(\frac{1}{\sqrt{2}}\right)$.

10.4. Biharmonic Submanifolds with Constant Mean and Scalar Curvatures

For a biharmonic hypersurface in a real space form ($R^6\left(c\right)$) with four distinct principal curvatures and a constant norm of the second fundamental form, R. S. Gupta proved the following.

Theorem 30

([66]). If M is a biharmonic hypersurface of $R^6\left(c\right)$ with, at most, four distinct principal curvatures and a constant norm of the second fundamental form, then M has constant mean curvature and scalar curvature.

11. PNMCV Biharmonic Submanifolds in Space Forms

11.1. PNMCV Biharmonic Submanifolds in Spheres

Balmuş, Montaldo, and Oniciuc studied PNMCV biharmonic submanifolds in spheres and proved the following.

Theorem 31

([67]). If $\phi :M\to S^m\left(1\right)$ is a biharmonic submanifold with a parallel normalized mean curvature vector and has, at most, two distinct principal curvatures in the direction of the mean curvature vector, then it has a parallel mean curvature vector.

Theorem 32

([67]). Let $\phi :M\to S^m\left(1\right)$ be a biharmonic submanifold with a parallel normalized mean curvature vector. If φ has, at most, two distinct principal curvatures in the direction of $\overrightarrow{H}$, then either φ induces a minimal immersion of M in $S^{m-1}\left(\frac{1}{\sqrt{2}}\right)$ or, locally,

$$\phi \left(M\right)=M_1^{n_1}\times M_2^{n_2}\subset S^{m_1}\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\times S^{m_2}\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\subset S^m\left(1\right),$$

where $M_i^{n_i}$ is a minimal embedded submanifold of $S^{m_i}\left(\frac{1}{\sqrt{2}}\right)$ with $i=1,2$, $n_1+n_2=dimM$, $n_1\ne n_2$, and $m_1+m_2=m-1$.

11.2. PNMCV Biharmonic Submanifolds in Pseudo-Riemannian Space Forms

In 2019, L. Du and J. Zhang [68] studied biharmonic submanifolds in a pseudo-Riemannian space form with a parallel normalized mean curvature vector field and classified such submanifolds.

Theorem 33

([68]). Let $M_r^n$ be a pseudo-Riemannian submanifold of a pseudo-Riemannian space form ($R_s^{n+p}\left(c\right)$) with a parallel normalized mean curvature vector field. Assume that $M_r^n$ has a diagonalizable shape operator with, at most, two distinct principal curvatures in the direction of $\overrightarrow{H}$. If $M_r^n$ is biharmonic, then $c\ne 0$. Moreover, we have the following:

(a)

If $c<0$, then $\overrightarrow{H}$ is time-like and $\langle \overrightarrow{H}.\overrightarrow{H}\rangle \le -c$. The equality holds if and only if $M_r^n$ is pseudo-umbilical;

(b)

If $c>0$, then $\overrightarrow{H}$ is space-like and $\langle \overrightarrow{H},\overrightarrow{H}\rangle >\le -c$. The equality holds if and only if $M_r^n$ is pseudo-umbilical.

From Theorem 33, we have the following.

Corollary 2

([68]). Let $\varphi :M^n\to {\mathbb{E}}^{n+2}$ be an isometric immersion of a Riemannian n manifold ($M^n$) into ${\mathbb{E}}^{n+2}$ with a parallel normalized mean curvature vector field. If ϕ has, at most, two distinct principal curvatures in the direction of $\overrightarrow{H}$, then ϕ is never biharmonic.

12. Biharmonic Hypersurfaces in Einstein Spaces

For biharmonic hypersurfaces in Einstein manifolds, Maeta and Ou proved the following.

Theorem 34

([63]). A compact Einstein hypersurface ($M^n$) of an Einstein manifold $\left((N^{n+1},\tilde{g})\right)$ with ${\mathrm{Ric}}^N=\lambda \tilde{g}$ is biharmonic if and only if it is either minimal or ${\parallel A\parallel }^2=\lambda$. Furthermore, in the latter case, the hypersurface has positive scalar curvature, i.e., $\tau =(n-2)\lambda +n^2{\left|\overrightarrow{H}\right|}^2>0$.

Theorem 35

([63]). A complete biharmonic hypersurface ($M^n$) of constant scalar curvature in a non-positively curved Einstein manifold $(N^{n+1},\tilde{g})$ is minimal if one of the following two cases occurs:

(i) 

$|\nabla \overrightarrow{H}|\in L^p$ for some $2<p<\infty$.

(ii) 

$|\nabla \overrightarrow{H}|\in L^2$ and the Ricci curvature of $M^n$ is bounded from below by $-c\{1+r^2\left(x\right)\}$, where $r\left(x\right)$ denotes the distance function on M.

13. ${\mathit{L}}_{\mathbf{1}}$-Biharmonic Hypersurfaces

13.1. $L_k$-Biharmonic Hypersurfaces and $L_k$ Conjecture

Let $M^n$ be a Riemannian (or pseudo-Riemannian) n manifold. The operators expressed as $L_k(k=0,1,\dots ,n-1)$ are extensions of the Laplacian expressed as $\Delta =L_0$ on $M^n$, representing the linearized operator of the first variation of the $(k+1)$-th $(k=0,1,\dots ,n-1)$ mean curvature functions (see, e.g., [69] or [27], pages 413–414 for more details).

Operator $L_k$ is expressed as

$$L_k\left(f\right)=\mathrm{Trace}(P_k\circ {\nabla }^{2}f),$$

(35)

where f is a smooth function ($M^n$), ${\nabla }^{2}f$ is the Hessian of f, and $P_k$ is the k-th Newton transformation associated with the second fundamental form of $M^n$.

In contrast to the usual Laplacian ($\Delta$, which is elliptic), the operators expressed as $L_r,r=1,\dots ,n,$ are not elliptic in general, but they still share some properties with the Laplacian of $M^n$. For instance, for an isometric immersion ($\varphi :M^n\to {\mathbb{E}}^{n+1}$), we have the following Beltrami-type formula (see [70]):

$$\begin{array}{c}\hfill L_r\varphi =-(n-r)\left({\displaystyle \genfrac{}{}{0pt}{}{n}{r}}\right)H_{r+1}\xi ,\end{array}$$

(36)

where $H_{r+1}$ is the $(r+1)$-th mean curvature and $\xi$ is a unit normal vector field of $\varphi$.

Definition 8.

A hypersurface ($\varphi :M_r^n\to {\mathbb{E}}_s^{n+1}$) is said to be $L_k$-biharmonic if it satisfies $L_k^2\varphi =0$ identically, and it is said to be $L_k$-biconservative if the tangent component of $L_k^2\varphi$ vanishes identically.

The following conjecture was made by M. Aminian and S. M. B. Kashani in 2015.

• ${\mathit{L}}_{\mathit{k}}$-conjecture ([71]). Every Euclidean hypersurface $(\varphi :M^n\to {\mathbb{E}}^{n+1})$ satisfying the condition of

  $$L_k^2\varphi =0$$

  (37)

  for some $k(0\le k\le n-1)$ is k-minimal, i.e., its $(k+1)$-th mean curvature $H_{k+1}$ vanishes identically.

13.2. $L_1$-Biharmonic Hypersurfaces in ${\mathbb{E}}^{n+1}$

In 2018, A. Mohammadpouri, F. Pashaie, and S. Tajbakhsh [72] proved the following.

Theorem 36.

Every $L_1$-biharmonic hypersurface in Euclidean space (${\mathbb{E}}^{n+1}$) with constant mean curvature and three distinct principal curvatures is 1-minimal.

Theorem 36 provides a partial solution to the $L_1$ version of Chen’s biharmonic conjecture, stating that every $L_1$-biharmonic hypersurface in ${\mathbb{E}}^{n+1}$ is 1-minimal.

In 2020, A. Mohammadpouri and F. Pashaie studied hypersurfaces in Euclidean space that satisfy the biharmonicity-like condition ($L_k^2\varphi =0$) for $k=1$ or $k=2$. They proved the following.

Theorem 37

([73]). Every $L_k$-biharmonic hypersurface with $k=1$ or $k=2$ in Euclidean 4-space (${\mathbb{E}}^4$) with a constant k-th mean curvature is k-minimal.

13.3. $L_1$-biharmonic Time-like Hypersurfaces in ${\mathbb{E}}_1^4$

In 2020, F. Pashaie [74] studied $L_1$-biharmonic time-like hypersurfaces in pseudo-Euclidean space. He proved the next three theorems.

Theorem 38

([74]). Every $L_1$-biharmonic, orientable, time-like hypersurface in Lorentzian 4-space (${\mathbb{E}}_1^4$) with constant mean curvature and three distinct principal curvatures is 1-minimal.

Theorem 39

([74]). Each $L_1$-biharmonic, orientable, time-like hypersurface in ${\mathbb{E}}_1^4$ with two distinct complex and one constant real principal curvatures and constant mean curvature is 1-minimal.

Theorem 40

([74]). Every $L_1$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with, at most, two distinct principal curvatures is 1-minimal.

F. Pashaie, N. Tanoomand-Khooshmehr, A. Rahimi, and L. Shahbaz also investigated the $L_1$ version of Chen’s biharmonic conjecture on time-like hypersurfaces in ${\mathbb{E}}_1^5$ with three distinct principal curvatures and constant mean curvature. They proved the following.

Theorem 41

([75]). Every $L_1$-biharmonic, orientable, Lorentzian hypersurface ($M_1^4$ in ${\mathbb{E}}_1^5$) with a diagonal shape operator, constant mean curvature, and non-constant second mean curvature has a non-constant principal curvature of multiplicity one.

Theorem 42

([75]). Let $\varphi :M_1^4\to {\mathbb{E}}_1^5$ be an $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a diagonal shape operator, three distinct principal curvatures, and constant mean curvature, which is 1-minimal.

Remark 5.

F. Pashaie et al. [75] also obtained similar results for $L_1$-biharmonic Lorentzian hypersurfaces of ${\mathbb{E}}_1^5$ with a non-diagonal shape operator.

13.4. $L_1$-Biharmonic Lorentzian Hypersurfaces in ${\mathbb{E}}_1^5$

The shape operator of a Lorentzian hypersurface ($M_1^4$) in ${\mathbb{E}}_1^5$ has four possible canonical matrix forms, namely types I, II, III, and IV (refer to [76,77] for details).

In 2023, F. Pashaie [78] studied Lorentzian hypersurfaces of ${\mathbb{E}}_1^5$ that have, at most, two different principal curves. He proved the following.

Theorem 43

([78]). Every proper $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a shape operator of type I cannot have a principal curvature of multiplicity four.

Theorem 44

([78]). If $\varphi :M^4\to {\mathbb{E}}_1^5$ is a proper $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a type I shape operator, then it cannot have two distinct principal curvatures of multiplicities 2 or 3 and 1.

Theorem 45

([78]). If $\varphi :M^4\to {\mathbb{E}}_1^5$ is an $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a shape operator of type II, at most two distinct principal curvatures, and constant ordinary mean curvature, then it is not proper.

Theorem 46

([78]). If $\varphi :M_1^4\to {\mathbb{E}}_1^5$ is an $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a shape operator of type III and constant ordinary mean curvature, then it is not proper.

Theorem 47

([78]). Every $L_1$-biharmonic Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a shape operator of type IV and, at most, two distinct non-zero principal curvatures has constant second mean curvature. Furthermore, such a hypersurface is minimal and 2-minimal.

Theorem 48

([78]). Let $\varphi :M_1^4\to {\mathbb{E}}_1^5$ be an $L_1$-biharmonic Lorentzian hypersurface of ${\mathbb{E}}_1^5$ with a diagonal shape operator, three distinct principal curvatures, and constant mean curvature, which is 1-minimal.

14. ${\mathit{L}}_{\mathit{k}}$-Biharmonic Hypersurfaces

14.1. $L_k$-Biharmonic Hypersurfaces in Euclidean Spaces

On the $L_k$ conjecture, Aminian and Kashani [71] proved the following.

Theorem 49

([71]). If $\varphi :M^n\to {\mathbb{E}}^{n+1}$ is of $L_k$-finite type and $L_k^2\varphi =0$, then $M^n$ is of null 1 type.

Theorem 50

([71]). Let $\varphi :M^n\to {\mathbb{E}}^{n+1}$ be a hypersurface satisfying $L_k^2\varphi =0$ for some $k,1\le k\le n-1$. Suppose that M has two principal curvatures, both with multiplicities greater than one. Then, $M^n$ is k-minimal, i.e., $H_{k+1}=0$.

As a consequence of Theorem 50, Aminian and Kashani obtained the following.

Theorem 51

([71]). The only $L_k$-biharmonic hypersurfaces in ${\mathbb{E}}^{n+1}$ with two principal curvatures, both with multiplicities greater than one, are the standard product embeddings $(S^p\left(r\right)\times {\mathbb{E}}^{n-p}\subset {\mathbb{E}}^{n+1})$ for $r>0$ and $n\le k$.

14.2. $L_k$-Biharmonic Hypersurfaces in Hyperbolic Spaces

M. Aminian and S. M. B. Kashani [79] studied $L_k$-biharmonic hypersurfaces in hyperbolic spaces and proved the following.

Theorem 52

([79]). Let $M^n$ be an oriented hypersurface of a hyperbolic $(n+1)$ space ($H^{n+1}$) such that it has two distinct principal curvatures with both multiplicities greater than one. If $M^n$ is $L_k$-biharmonic, then it is k-minimal.

Theorem 53

([79]). There is no $L_k$-biharmonic oriented hypersurface in $H^{n+1}$ that has two distinct principal curvatures with both multiplicities greater than one.

Theorem 54

([79]). Let $M^n$ be an oriented hypersurface of $H^{n+1}$ whose multiplicities of its principal curvatures of $M^n$ are 1 and $n-1$. If $M^4$ is $L_k$-biharmonic, then it is k-minimal.

Theorem 55

([79]). The $L_k$ conjecture is true for any weakly convex $L_k$-biharmonic hypersurface ($M^n$) of ${\mathbb{E}}^{n+1}$ or $H^{n+1}$.

Remark 6.

Theorem 55 extended Theorem 6.

14.3. $L_k$-Biharmonic Hypersurfaces in Real Space Forms $R^4\left(c\right)$

In 2020, M. Aminian investigated $L_k$-biharmonic hypersurfaces in four-dimensional real space forms with three distinct principal curvatures. He obtained the following.

Theorem 56

([80]). Let $M^3$ be an oriented hypersurface of a real space form ($R^4\left(c\right)$) with three distinct principal curvatures. If $M^3$ is proper $L_2$-biharmonic and $H_2$ is constant, then $c=1$ and $M^3$ is congruent to an open portion of a tube of radius ${\displaystyle \frac{\pi }{12}}$ around the standard embedding of a real projective plane $R{P}^2$ into $S^4\left(1\right)$, and principal curvatures of $M^3$ are $2+\sqrt{3},2-\sqrt{3},-1$.

14.4. $L_k$-Biharmonic Hypersurfaces in ${\mathbb{E}}_1^4$

In 2020, F. Pashaie studied $L_k$-biharmonic Lorentzian hypersurfaces in ${\mathbb{E}}_1^4$. He proved the following.

Theorem 57

([81]). Let k be a natural number $\le 3$. Then, there is no Lorentzian $L_k$-biharmonic hypersurface of $L_k$-finite type in ${\mathbb{E}}_1^4$.

Theorem 58

([81]). Let k be a natural number $\le 3$ and $M^3$ be an $L_k$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with constant k-th mean curvature and a shape operator of type I with three distinct real principal curvatures, which is k-minimal.

Theorem 59

([81]). Let k be a natural number $\le 3$ and $M_1^3$ be an $L_k$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with constant k-th mean curvature and a shape operator of type I with has exactly two distinct principal curvatures and is k-minimal.

Theorem 60

([81]). Let k be a natural number $\le 3$ and $M_1^3$ be an $L_k$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with constant k-th mean curvature and a shape operator of type II with exactly two distinct principal curvatures, which is k-minimal.

Theorem 61

([81]). Let k be a natural number $\le 3$ and $M_1^3$ be an $L_k$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with a shape operator of type III with constant k-th mean curvature, which is k-minimal. Furthermore, all of the mean curvatures of $M_1^3$ are null.

Theorem 62

([81]). Let k be a natural number $\le 3$ and $M_1^3$ be an $L_k$-biharmonic, orientable, time-like hypersurface of ${\mathbb{E}}_1^4$ with a shape operator of type IV with constant k-th mean curvature and a constant real principal curvature, which is k-minimal.

15. Chen’s Biharmonic Conjecture and Chen’s Flow

In this section, we assume that $M^n$ is a complete Riemannian n manifold and $\phi :M^n\to {\mathbb{E}}^m$ is an isometric immersion into ${\mathbb{E}}^m$. Since $M^n$ is a biharmonic submanifold if the immersion ($\phi :M^n\to {\mathbb{E}}^m$) satisfies ${\Delta }^2\phi =0$, Y. Bernard, G. Wheeler, and V. Wheeler [82] considered the heat flow for ${\Delta }^2$, which is a one-parameter family of isometric immersions ($\phi :M^n\times [0,T)\to {\mathbb{E}}^m$) that satisfies the initial condition of $\phi (p,0)={\phi }_0\left(p\right)$ for a given isometric immersion (${\phi }_0:M^n\to {\mathbb{E}}^m$) such that

$${\displaystyle \frac{\partial \phi (p,t)}{\partial t}}=-\left({\Delta }^2\phi \right)(p,t)\forall (p,t)\in M^n\times (0,T).$$

(38)

Bernard et al. [82] named this fourth-order curvature flow Chen’s flow.

Since ${\Delta }^2$ is a fourth-order quasilinear elliptic operator, local existence and uniqueness for (38) are standard. Thus, we have the following:

Theorem 63.

If ${\phi }_0:M\to {\mathbb{E}}^m$ is a closed, isometrically immersed submanifold, then there is a finite time ($T\in (0,\infty ]$) and a unique one-parameter family of smooth, closed isometric immersions ($f:M^n\times [0,T)\to {\mathbb{E}}^m$) such that (38) is satisfied and T is maximal.

Since there are no closed, biharmonic submanifolds in Euclidean spaces, one expects that the flow exists only for, at most, a finite time, i.e., $T<\infty$.

Bernard, Wheeler, and Wheeler [82] proved the following theorem, which provides a precise estimate and is sharp for $n\in \{2,3,4\}$.

Theorem 64.

For a natural number $n\in \{2,3,4\}$, Chen’s flow ($\phi :M\times [0,T)\to {\mathbb{E}}^m$) with closed, smooth initial data (${\phi }_0:M\to {\mathbb{E}}^m$) has a finite maximal time of existence, with an explicit estimate of

$$T\le {\displaystyle \frac{\mu {\left({\phi }_0\right)}^{\frac{4}{n}}}{C_n}},$$

(39)

where $C_n=4{\omega }_n^{\frac{4}{n}}{n}^2$ and $C_n={\omega }_n^{\frac{4}{n}}/\left(n^2{4}^{4n+3}\right)$ for $n>4$ and ${\omega }_n$ is the area of the unit n sphere. Moreover, one has $\mu \left({\phi }_t\right)\searrow 0$ as $t\nearrow T$ whenever the equality in (39) is achieved.

Remark 7.

From this theorem, if follows that round spheres are driven to points under Chen’s flow with $T=\frac{r_0^4}{4n^2}$, where $r_0$ is the initial radius. Consequently, the estimate (39) is sharp when the dimension of M is $2,3$, or 4.

Remark 8.

For further results on Chen’s flows, we refer readers to [83,84].

16. $\mathit{H}$-Hypersurfaces and Biconservative Submanifolds

The notion of H submanifolds of Euclidean spaces was defined in 1995 by T. Hasanis and T. Vlachos in [8], requiring merely the vanishing of the tangential component of ${\Delta }^2\varphi$ instead of ${\Delta }^2\varphi =0$; the authors studied H-hypersurfaces in ${\mathbb{E}}^3$ and ${\mathbb{E}}^4$. Later, H-submanifolds were also named biconservative submanifolds by Caddeo, Montaldo, Oniciuc, and Piu (see [85] for details).

The concept of biconservative submanifolds is derived from the notion of biharmonic submanifolds, requiring only the vanishing of the tangential component of ${\Delta }^2\varphi$ for Euclidean submanifolds or the tangential component of ${\tau }_2\left(\phi \right)$ for Riemannian submanifolds.

An advantage of investigating biconservative submanifolds is that one may obtain information on the influence of the tangential component of the biharmonic equation on submanifolds. Therefore, it may provide us with information on biharmonic conjectures.

16.1. H-Hypersurfaces in Euclidean Spaces

In 1995, T. Hasanis and T. Vlachos [8] proved the following classification theorem for H-hypersurfaces in ${\mathbb{E}}^4$.

Theorem 65.

If M is a biconservative hypersurface of ${\mathbb{E}}^4$, then M is one of the following hypersurfaces:

(a)

A hypersurface with constant mean curvature (i.e., CMC);

(b)

A rotational hypersurface with constant mean curvature generated by a unit speed plane curve ($\left(f\right(s),g(s\left)\right)$, where f satisfies second-order ordinary differential equation $3f{f}^{″}=2(1-{\left(f^{\prime }\right)}^2)$);

(c)

A generalized cylinder on a surface of revolution in ${\mathbb{E}}^3$ with non-constant mean curvature parameterized by

$$x(u,v,s)=\left(f\left(s\right)cosu,f\left(s\right)sinu,v,g\left(s\right)\right),$$

such that $\left(f\right(s),g(s\left)\right)$ defines a unit speed curve with non-constant curvature and f is a solution of second-order ordinary differential equation $3f{f}^{″}=1-{\left(f^{\prime }\right)}^2$;

(d)

A $SO\left(2\right)\times SO\left(2\right)$-invariant hypersurface of ${\mathbb{E}}^4$ defined by

$$\left(f\left(s\right)cosu,f\left(s\right)sinu,g\left(s\right)cosv,g\left(s\right)sinv\right),$$

with non-constant mean curvature, where $\left(f\right(s),g(s\left)\right)$ defines a unit speed curve with non-constant curvature and f and g are smooth functions satisfy differential equation

$$f^{\prime }{g}^{″}-f^{″}{g}^{\prime }={\displaystyle \frac{1}{3}}\left({\displaystyle \frac{f^{\prime }}{g}}-{\displaystyle \frac{g^{\prime }}{f}}\right).$$

16.2. Biconservative Submanifolds and H-Submanifolds

According to David Hilbert [86], the stress–energy tensor associated with a variational problem is a symmetric two-covariant tensor ($\widehat{S}$) that satisfies $\mathrm{div}\widehat{S}=0$ at critical points; in other words, $\widehat{S}$ is a conservative tensor. In the context of harmonic maps ($\varphi :(M,g)\to (\tilde{M},\tilde{g})$) that are critical points of energy, the stress–energy tensor associated with the critical point of $E\left(\varphi \right)=\frac{1}{2}{\int }_M{\left|d\varphi \right|}^2\mathrm{d}{v}_g,$ was studied in detail in [87] by P. Baird and J. Eells and [88] by A. Sanini. In fact, the Euler–Lagrange equation for energy is equivalent to the vanishing of the tension field (${\tau }_1\left(\varphi \right)$) (cf. [31]), and the tensor

$$\widehat{S}={\displaystyle \frac{1}{2}}{\left|d\varphi \right|}^{2}g-{\varphi }^{*}\tilde{g}$$

satisfies $\mathrm{div}\widehat{S}=-\langle {\tau }_1\left(\varphi \right),d\varphi \rangle$. Therefore, we have $\mathrm{div}\widehat{S}=0$ if the map is harmonic. Furthermore, for isometric immersions, the condition of $\mathrm{div}\widehat{S}=0$ is always satisfied because ${\tau }_1\left(\varphi \right)$ is normal.

The study of the stress–energy tensor for bi-energy was studied by Jiang in [1] and further developed in [89]. Its expression is given by

$$\begin{array}{ccc}\hfill {\widehat{S}}_2(X,Y)& =& {\displaystyle \frac{1}{2}}{\left|{\tau }_1\left(\varphi \right)\right|}^2\langle X,Y\rangle +\langle d\varphi ,\nabla {\tau }_1\left(\varphi \right)\rangle \langle X,Y\rangle -\langle d\varphi \left(X\right),{\nabla }_Y{\tau }_1\left(\varphi \right)\rangle -\langle d\varphi \left(Y\right),{\nabla }_X{\tau }_1\left(\varphi \right)\rangle ,\hfill \end{array}$$

which satisfies

$$\mathrm{div}{\widehat{S}}_2=-\langle {\tau }_2\left(\varphi \right),d\varphi \rangle ,$$

(40)

conforming to the principle of a stress–energy tensor for the bi-energy.

Let $\varphi :(M,g)\to (\tilde{M},\tilde{g})$ be an isometric immersion. Then, (40) becomes

$$\begin{array}{c}\hfill \mathrm{div}{\widehat{S}}_2=-{\tau }_2{\left(\varphi \right)}^T.\end{array}$$

(41)

Applying these facts, Caddeo, Montaldo, Oniciuc, and Piu [85] defined biconservative submanifolds of a Riemannian manifold as submanifolds that satisfy the condition of $\mathrm{div}{\widehat{S}}_2=0$. As a fact, a biconservative submanifold of an Euclidean space is nothing but an H-submanifold in the sense of T. Hasanis and T. Vlachos proposed in [8].

17. Biconservative Submanifolds in Euclidean and Lorentzian Spaces

17.1. Biconservative Hypersurfaces in ${\mathbb{E}}^5$

In 2019, R. S. Gupta [90] studied biconservative hypersurfaces in ${\mathbb{E}}^5$. He proved the following.

Theorem 66

([90]). Any biconservative hypersurface of ${\mathbb{E}}^5$ with a constant norm of the second fundamental form must have constant mean curvature.

Theorem 66 implies the following two corollaries.

Corollary 3.

Any null 2-type hypersurface of ${\mathbb{E}}^5$ with a constant norm of the second fundamental form must have constant mean curvature.

Corollary 4.

Every biharmonic hypersurface of ${\mathbb{E}}^5$ with a constant norm of the second fundamental form is minimal.

17.2. Ideal Biconservative Hypersurfaces in Euclidean Spaces

In 2013, the author and Munteanu established the following.

Theorem 67

([50]). Any biconservative $\delta \left(2\right)$-ideal hypersurface in ${\mathbb{E}}^{n+1}(n\ge 3)$ must have constant mean curvature.

In 2018, Deepika and A. Arvanitoyeorgos improved Theorem 67 as follows.

Theorem 68

([91]). Any biconservative $\delta \left(2\right)$-ideal hypersurface in ${\mathbb{E}}^{n+1}$ must be minimal.

Theorem 69

([91]). Any biconservative $\delta \left(3\right)$-ideal hypersurface in ${\mathbb{E}}^5$ has constant mean curvature.

Theorem 70

([91]). Any biconservative $\delta \left(4\right)$-ideal hypersurface in ${\mathbb{E}}^5$ with constant scalar curvature must have constant mean curvature.

In 2020, Deepika and A. Arvanitoyeorgos proved the following.

Theorem 71

([51]). Every biconservative $\delta \left(r\right)$-ideal hypersurface in ${\mathbb{E}}^{n+1}(n\ge 3)$ with, at most, $r+1$ distinct principal curvatures must have constant mean curvature.

17.3. Holonomic Biconservative Hypersurfaces of Euclidean Space

Recently, H. Bibi, M Soret, and M. Ville [42] studied and constructed biconservative hypersurfaces (M) of a Euclidean space that are holonomic. They proved the following results.

Theorem 72

([42]). A holonomic proper biconservative hypersurface ($M^n$) in ${\mathbb{E}}^{n+1}$ s foliated by the level sets of the mean curvature function (α) of $M^n$, which are isoparametric codimension 2 submanifolds of ${\mathbb{E}}^{n+1}$ in a neighborhood of a regular point of α. The symmetries of the leaves extend to symmetries of $M^n$. The integral curves of the gradient field ($\nabla \alpha$) on $M^n$ are congruent planar curves governed by an ODE of order two.

Conversely, for any codimension 2 isoparametric submanifold ($U_0\subset {\mathbb{E}}^{n+1}$) that is holonomic, there locally exists a proper biconservative hypersurface obtained by a local normal evolution of $U_0$ in the normal bundle of $U_0$ that preserves the fibers of the normal bundle of $U_0$ and the symmetry group of $U_0$.

Theorem 72 implies the following.

Corollary 5.

Holonomic proper biconservative hypersurfaces of ${\mathbb{E}}^{n+1}$ are foliated by either round spheres ($S^{n-1}$); products of round spheres ($S^p\times S^q$); cylinders ($S^p\times {\mathbb{E}}^q)$, where $p+q=n-1$; or $(S^p\times S^q\times {\mathbb{E}}^r)$, where $p+q+r=n-1$.

17.4. PNMCV Biconservative Submanifold of Codimension Two in ${\mathbb{E}}^{n+2}$

In [46], R. Y. Şen and N. C. Turgay studied biconservative, non-CMC surfaces with a flat, normal bundle in ${\mathbb{E}}^4$. First, they locally determined all biconservative surfaces in ${\mathbb{E}}^4$ with a parallel normalized mean curvature vector by showing that they are rotational surfaces. Then, they considered meridian surfaces lying on a rotational hypersurface of ${\mathbb{E}}^4$. After that, they constructed a family of biconservative surfaces with a flat, normal bundle and a non-parallel normalized mean curvature vector. In particular, they proved the following theorem, among others.

Theorem 73

([46]). If M is a proper biconservative PNMCV surface in ${\mathbb{E}}^4$, then locally, it is congruent to the rotational surface parameterized by

$$x(s,t)=\left({\alpha }_1\left(s\right)cost,{\alpha }_1\left(s\right)sint,{\alpha }_2\left(s\right),{\alpha }_3\left(s\right)\right)$$

(42)

with an arc-length smooth profile curve of $\alpha \left(s\right)=({\alpha }_1\left(s\right),{\alpha }_1\left(s\right),{\alpha }_1\left(s\right))$ and

$${\alpha }_1\left(s\right)={\displaystyle \frac{1}{c_{2}f{\left(s\right)}^{3/4}}},$$

whose curvature and torsion functions are given by

$$\kappa \left(s\right)=f\left(s\right)\sqrt{1+c^{2}f\left(s\right)},\tau \left(s\right)={\displaystyle \frac{c{f}^{\prime }\left(s\right)}{2\sqrt{f\left(s\right)}(1+c^{2}f\left(s\right))}}.$$

In 2022, R. Y. Şen [92] studied biconservative n-dimensional submanifolds ($M^n$) in ${\mathbb{E}}^{n+2}$ that have a parallel normalized mean curvature vector field. He also determined canonical forms of the shape operators for such submanifolds.

17.5. Biconservative Lorentzian Hypersurfaces in ${\mathbb{E}}_1^{n+1}$

In 2019, Gupta and Sharfuddin studied biconservative Lorentz hypersurfaces ($M_1^n$) of ${\mathbb{E}}_1^{n_{+}1}$ whose shape operators have complex eigenvalues. They obtained the following.

Theorem 74

([93]). If $M_1^n$ is a biconservative Lorentz hypersurface in ${\mathbb{E}}_1^{n_{+}1}$ with complex eigenvalues, then it must have constant mean curvature.

Theorem 75

([93]). Any biharmonic Lorentz hypersurface ($M_1^n$) in ${\mathbb{E}}_1^{n_{+}1}$ with complex eigenvalues is minimal.

17.6. PNMCV Biconservative Hypersurfaces in ${\mathbb{E}}_1^5$

In 2024, R. Y. Şen [94] investigated biconservative submanifolds in ${\mathbb{E}}_1^5$ with a parallel normalized mean curvature vector field. She obtained explicit classifications for the biconservative PNMCV submanifolds with two distinct principal curvatures of the shape operator in the direction of the mean curvature vector field. He also investigated such submanifolds in ${\mathbb{E}}_1^5$ with time-like or space-like mean curvature vector fields.

18. Biconservative Hypersurfaces in Real and Lorentzian Space Forms

In 2022, complete biconservative, non-CMC surfaces in a real space form ($R^3\left(c\right)$) were completely classified by Nistor and Oniciuc in [95] (see also [96], subsection 5.6). Now, let us discuss biconservative hypersurfaces in a real space form ($R^{n+1}\left(c\right)$) with $n>3$.

18.1. Biconservative Hypersurfaces in $S^4$ and $H^4$ with Three Distinct Principal Curvatures

In 2019, Turgay and Upadhyay [97] investigated biconservative hypersurfaces in $S^n$ and $H^n$. In particular, they proved the next two interesting local classification theorems for biconservative hypersurfaces in the sphere ($S^4$) and hyperbolic space ($H^4$).

Theorem 76

([97]). If M is a hypersurface in $S^4\left(1\right)$ with three distinct principal curvatures and a diagonalizable shape operator, then M is biconservative if and only if it is congruent to the submanifold in ${\mathbb{E}}^5$ defined by

$$x(s,t,u)=\left({\alpha }_1\left(s\right),{\alpha }_2\left(s\right)cost,{\alpha }_2\left(s\right)sint,{\alpha }_3\left(s\right)cost,{\alpha }_3\left(s\right)sint\right),$$

(43)

for an arc-length parameterized curve ($\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3):(a,b)\to S^2\left(1\right)$) with spherical curvature satisfying

$$\begin{array}{ccc}\hfill {\kappa }_S& =& {\displaystyle \frac{-3H}{2}},\hfill \end{array}$$

(44)

where $H=H\left(s\right)$ denotes the mean curvature function.

Theorem 77

([97]). A biconservative hypersurface (M) in $H^4(-1)$ with three distinct principal curvatures is congruent to one of the four hypersurfaces given below.

(1)

A hypersurface in $H^4(-1)$ defined by Equation (43) with an arc-length parameterized curve

$$\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3):(a,b)\to H^2(-1)).$$

(2)

A hypersurface in $H^4(-1)$ defined by

$$x(s,t,u)=\left({\alpha }_1\left(s\right)cosht,{\alpha }_1\left(s\right)sinht,{\alpha }_2\left(s\right)cost,{\alpha }_2\left(s\right)sint,{\alpha }_3\left(s\right)\right)$$

(45)

with an arc-length parameterized curve $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3):(a,b)\to H^2(-1)$.

(3)

A hypersurface in $H^4(-1)$ defined by

$$\begin{array}{cc}\hfill x(s,t,u)=& \left({\displaystyle \frac{aA{\left(s\right)}^2+a}{s}}+as{u}^2+{\displaystyle \frac{s}{4a}},su,A\left(s\right)cost,A\left(s\right)sint,{\displaystyle \frac{aA{\left(s\right)}^2+a}{s}}+as{u}^2-{\displaystyle \frac{s}{4a}}\right)\hfill \end{array}$$

(46)

with functions ${\alpha }_2$ and ${\alpha }_3$ and some non-zero constants (a and ${\alpha }_1$).

(4)

A hypersurface in $H^4(-1)$ is defined by

$$\begin{array}{cc}\hfill x(s,t,u)=& \left({\displaystyle \frac{aA{\left(s\right)}^2}{s}}+as\left(t^2+u^2\right)+{\displaystyle \frac{s}{4a}}+{\displaystyle \frac{a}{s}},st,su,A\left(s\right),{\displaystyle \frac{aA{\left(s\right)}^2}{s}}+as\left(t^2+u^2\right)-{\displaystyle \frac{s}{4a}}+{\displaystyle \frac{a}{s}}\right),\hfill \end{array}$$

(47)

with a function (A) and a non-zero constant (a).

18.2. Biconservative Hypersurfaces in $R^{n+1}\left(c\right)$ with 3 or 4 Distinct Principal Curvatures

In 2023, Y. Fu and X. Zhan [98] studied biconservative hypersurfaces of a real space form with three distinct principal curvatures. They obtained the following.

Theorem 78

([98]). If $M^n(n>4)$ is a biconservative hypersurface in a space form $\left(R^{n+1}\left(c\right)\right)$ with constant scalar curvature and with three distinct principal curvatures, then it is either a constant mean curvature hypersurface or it is contained in a certain non-CMC generalized cylinder $({\mathsf{\Sigma }}^4\times {\mathbb{E}}^{n-4})$, where ${\mathsf{\Sigma }}^4$ is a non-CMC rotational hypersurface in a ${\mathbb{E}}^5$.

Gupta and Arvanitoyeorgos [99] investigated biconservative hypersurfaces in a real space form with, at most, four distinct principal curvatures. They obtained the following.

Theorem 79

([99]). Let M be a biconservative hypersurface of a space form $\left({\overline{M}}^{n+1}\left(c\right)\right)$. If M has, at most, four distinct principal curvatures and its second fundamental form has a constant norm, then it has constant mean curvature and scalar curvature.

Remark 9.

For rigidity results on compact biconservative hypersurfaces in real space forms, we refer to [100,101,102].

18.3. Biconservative CMC Surfaces in 4D Lorentzian Space Forms

Classifications of biconservative surfaces in 4D Lorentzian space forms with constant mean curvature were archived by A. Kayhan and N. C. Turgay [103] as follows.

Theorem 80

([103]). A surface (M) in ${\mathbb{E}}_1^4$ with non-zero constant mean curvature is biconservative if and only if it is locally congruent to one of the following four types of surfaces.

(a)

A surface with a parallel mean curvature vector.

(b)

A ruled surface parameterized by $x(s,t)=\beta \left(s\right)+t\gamma \left(s\right)$ that has constant curvature (c) and is proper biconservative if β and γ satisfy

$$\langle \gamma ,\gamma \rangle =0,\langle {\gamma }^{\prime },{\gamma }^{\prime }\rangle =c^2,\langle {\beta }^{\prime },\gamma \rangle =-1,$$

where β is a curve and γ is a vector-valued function.

(c)

A cylinder parameterized by $x(s,t)=({\alpha }_1\left(s\right),{\alpha }_2\left(s\right),{\alpha }_3\left(s\right),t)$ for a unit speed curve ($\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)$) in ${\mathbb{E}}_1^3$ with a non-null normal vector field.

(d)

A cylinder parameterized by $x(s,t)=(t,{\alpha }_1\left(s\right),{\alpha }_2\left(s\right),{\alpha }_3\left(s\right))$ for a unit speed curve ($\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)$) in ${\mathbb{E}}^3$.

Theorem 81

([103]). If M is a surface in a non-flat Lorentzian space form ($R_1^4\left(\delta \right),\delta =\pm 1$), then M has non-zero constant mean curvature, and it is proper biconservative if and only if it is locally congruent to the ruled surface defined by $x(s,t)=\beta \left(s\right)+t\gamma \left(s\right)$ for some $\beta ,\gamma$ satisfying

$$\begin{array}{cc}& \langle \beta ,\beta \rangle =\delta ,\langle \beta ,\gamma \rangle =0,\langle \gamma ,\gamma \rangle =0,\hfill \\ & \langle \gamma ,\gamma \rangle =0,\langle {\gamma }^{\prime },{\gamma }^{\prime }\rangle =1+c^2,\langle {\beta }^{\prime },\gamma \rangle =-1,\hfill \end{array}$$

where c is the mean curvature of M.

The following two results follow easily from Theorem 81.

Theorem 82

([103]). If M is a proper biconservative surface with non-zero constant mean curvature (c) in the de Sitter space $\left(S_1^4\right)$, then M is biharmonic if and only if we have $c=1$.

Theorem 83

([103]). There do not exist proper biharmonic surfaces with non-zero constant mean curvature in the anti-de Sitter space $\left(H^4(-1)\right)$.

19. Biconservative Submanifolds in Complex Space Forms

In 2023, H. Bibi, B.-Y. Chen, D. Fetcu, and C. Oniciuc [104] studied parallel mean curvature submanifolds of complex space forms. They established the following results.

Theorem 84

([104]). Let $M^n$ be a submanifold in a complex space form (${\tilde{M}}^m\left(c\right)$). If $M^n$ has a parallel mean curvature vector, then we have the following:

(1)

For $c=0$, $M^n$ is a biconservative submanifold.

(2)

For $c\ne 0$, $M^n$ is a biconservative submanifold if and only if $J{\left(J\overrightarrow{H}\right)}^T\in \mathfrak{X}\left(T^{\perp }{M}^n\right)$.

Theorem 85

([104]). Let $M^n$ be a submanifold of complex space form (${\tilde{M}}^m\left(c\right)(c\ne 0)$) with a parallel mean curvature vector. If $J\overrightarrow{H}\in \mathfrak{X}\left(T^{\perp }{M}^n\right)$ or $J\overrightarrow{H}\in \mathfrak{X}\left(T{M}^n\right)$, then $M^n$ is a biconservative submanifold.

Theorem 86

([104]). If $M^2$ is a surface in a complex space form (${\tilde{M}}^m\left(c\right)$) with a parallel mean curvature vector, then we have

(1)

For $c=0$, $M^2$ is a biconservative surface.

(2)

For $c\ne 0$, $M^2$ is a biconservative surface if and only if $M^2$ is a totally real surface.

Theorem 86 implies the next two corollaries.

Corollary 6

([104]). Let $M^n$ be a totally real submanifold with a parallel mean curvature vector in a complex space form. Then, $M^n$ is a biconservative submanifold.

Corollary 7

([104]). Any real hypersurface of a complex space form (${\tilde{M}}^n\left(c\right)$) with a parallel mean curvature vector is biconservative.

For slant surfaces, Theorem 86 implies the following.

Corollary 8

([104]). Every proper slant surface with a parallel mean curvature vector in a non-flat complex space form is never a biconservative surface.

20. ${\mathit{L}}_{\mathit{k}}$-Biconservative Hypersurfaces in Lorentzian Spaces

20.1. $L_k$-Biconservative, Time-like Hypersurface in ${\mathbb{E}}_1^4$

In 2022, F. Pashaie [105] studied $L_k$-biconservative, time-like hypersurface in Lorentzian 4-space (${\mathbb{E}}_1^4$). He proved the following.

Theorem 87

([105]). Let $M_1^3$ be a time-like, $L_k$-biconservative hypersurface of ${\mathbb{E}}_1^4$ with a diagonalizable shape operator with constant k-th mean curvature and exactly two distinct principal curvatures (for $k<3$). Then, $M_1^3$ has constant $(k+1)$-th mean curvature.

Theorem 88

([105]). For $k<3$, if $M_1^3$ is a time-like, $L_k$-biconservative hypersurface in ${\mathbb{E}}_1^4$ with a shape operator of type II with exactly two distinct principal curvatures and constant k-th mean curvature, then $M_1^3$ has constant $(k+1)$-th mean curvature.

Theorem 89

([105]). For $k<3$, if $M_1^3$ is a time-like, $L_k$-biconservative hypersurface in ${\mathbb{E}}_1^4$ with a shape operator of type III with exactly two distinct principal curvatures and constant k-th mean curvature, then $M_1^3$ has constant $(k+1)$-th mean curvature.

Theorem 90

([105]). For $k<3$, if $M_1^3$ is a time-like, $L_k$-biconservative hypersurface in ${\mathbb{E}}_1^4$ with with a shape operator of type IV with constant k-th mean curvature and constant real principal curvature, then the second and third mean curvatures of $M_1^3$ are constant.

20.2. $L_k$-Biconservative Lorentzian Hypersurface in ${\mathbb{E}}_1^{n+1}$

Pashaie [106] studied $L_k$-biconservative Lorentzian hypersurfaces in Lorentzian 5-space. In particular, he proved the following (see also [107]).

Theorem 91

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a diagonalizable shape operator and a principal curvature of multiplicity four, then $M_1^4$ has constant $(k+1)$-th mean curvature.

Theorem 92

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a diagonalizable shape operator with constant ordinary mean curvature and exactly two principal curvatures of multiplicities 3 and 1 (respectively), then $M_1^4$ has constant $(k+1)$-th mean curvature.

Theorem 93

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a diagonalizable shape operator and two principal curvatures (λ and η, both of multiplicity 2), then $M_1^4$ has constant $(k+1)$-th mean curvature.

Theorem 94

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a shape operator of type II and, at most, two principal curvatures and constant ordinary mean curvature, then $M_1^4$ has constant $(k+1)$-th mean curvature.

Theorem 95

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a shape operator of type III and constant ordinary mean curvature, then $M_1^4$ has constant $(k+1)$-th mean curvature.

Theorem 96

([106]). For $k<4$, if $M_1^4$ is an $L_k$-biconservative Lorentzian hypersurface in ${\mathbb{E}}_1^5$ with a shape operator of type IV and, at most, two distinct non-zero principal curvatures, then $M_1^4$ has constant $(k+1)$-th mean curvature and is isoparametric.

For $k=1$, F. Pashaie [108] extended Theorems 91–96 to biconservative Lorentzian hypersurfaces in ${\mathbb{E}}_1^5$ and tobiconservative Lorentzian hypersurfaces (${\mathbb{E}}_1^{n+1}$) with $n\ge 4$.

Remark 10.

For further results on biconservative submanifolds, we refer reared to recent survey articles [96,102].

21. Triharmonic Hypersurfaces

Recall that, in 1964, Eells and Sampson [31] extended harmonic maps to polyharmonic maps of higher order (k), i.e., critical points of the k-energy functional ($E_k\left(\varphi \right)$). In 1986, Jiang [17] studied the first and second variational formulas of $E_2$. The first and second variational formulas of $E_k$ with $k\ge 3$ were derived independently by Wang [109] in 1989 and by Maeta [110] in 2012, respectively. Polyharmonic maps were also rigorously studied in many papers (cf., e.g., [110,111,112,113,114,115,116] and the references therein.) Triharmonic maps are called triharmonic submanifolds if the maps are isometric immersions. During the last two decades or so, there has been a growing interest in triharmonic maps and triharmonic submanifolds.

The two basic problems in triharmonic submanifolds are the minimality problem and the classification problem. The minimality problem is one of the main themes in this field, since minimal submanifolds are triharmonic, and many interesting results have been obtained.

In 2012, Maeta [116] extended Chen’s biharmonic conjecture to the next conjecture for k-harmonic submanifolds.

• k-Harmonic Conjecture: For any integer ($k\ge 3$), k-harmonic submanifolds in ${\mathbb{E}}^{n+1}$ are minimal.

In [116], Maeta pointed out that this conjecture is true for compact triharmonic CMC hypersurfaces; namely, he pointed out the following.

Theorem 97

([116]). Any compact CMC triharmonic hypersurface in $R^{n+1}\left(c\right)(c\le 0)$ is minimal.

21.1. Triharmonic CMC Hypersurfaces in Space Forms

In 2023, H. Chen and Z. Guan [117] studied non-compact, CMC, proper triharmonic hypersurfaces in a real space form ($R^{n+1}\left(c\right)$) with, at most, three distinct principal curvatures. They obtained the following.

Theorem 98

([117]). If $M^n$ is a CMC proper triharmonic hypersurface of a real space form ($R^{n+1}\left(c\right)(n\ge 3)$) with, at most, three distinct principal curvatures, then $M^n$ has constant scalar curvature.

Since the constancy of scalar curvature of a CMC hypersurface in the real space form is equivalent to the constancy of the squared norm of the second fundamental form, an immediate consequence of Theorem 97 is the following.

Corollary 9

([117]). If $M^n(n\ge 3)$ is a CMC triharmonic hypersurface of $R^{n+1}\left(c\right)(c\le 0)$ with, at most, three distinct principal curvatures, then it is minimal.

For $c=1$, there are explicit examples of non-minimal, k-harmonic hypersurfaces in a sphere. The typical examples are isoparametric ones—for instance, a small sphere ($S^n\left(r\right)(0<r<1)$) and the generalized Clifford torus defined by

$$S^m\left(r\right)\times S^{n-m}\left(\sqrt{1-r^2}\right)(0<r<1).$$

In order for $S^n\left(r\right)$ or $S^m\left(r\right)\times S^{n-m}\left(\sqrt{1-r^2}\right)$ to become k-harmonic, there are restrictions on parameters r and m.

Comparing Theorem 29, Chen and Guan imposed a sharp upper bound for a hypersurface that is not totally umbilical and satisfies the condition given in Theorem 103. In fact, Chen and Guan obtained the following.

Theorem 99

([117]). Let $M^n$ be a proper, CMC, k-harmonic ($k\ge 3$) hypersurface in $S^{n+1}(n\ge 3)$ with constant scalar curvature. If $M^n$ is not totally umbilical in $S^{n+1}$, then we have $|\overrightarrow{H}{|}^2\in (0,t_0]$, where $t_0$ is the largest real root of the polynomial:

$$\begin{array}{cc}\hfill f_{n,k}\left(t\right)& =n^4{t}^3+n^2\left\{(1-k)n^2+2(2+k)n-2(k+2)\right\}{t}^2\hfill \\ \hfill & +(1-n)\left\{3n-(k+2)\right)\left((k-1)n-k-2)\right\}t+(1-n){(n-2)}^2.\hfill \end{array}$$

Furthermore, $|\overrightarrow{H}{|}^2=t_0$ holds if and only if $M^n$ is locally $S^{n-1}\left(r\right)\times S^1\left(\sqrt{1-r^2}\right)$ with

$$r^2={\displaystyle \frac{2{(n-1)}^2}{n^2|\overrightarrow{H}{|}^2+2n(n-1)+n|\overrightarrow{H}|\sqrt{n^2{|\overrightarrow{H}|}^2+4(n-1)}}}.$$

(48)

The next three corollaries were also proven by Chen and Guan [117].

Corollary 10

([117]). Let $M^n$ be a proper, CMC triharmonic hypersurface in $S^{n+1}(n\ge 3)$ with, at most, three distinct principal curvatures. Then, we have one of the following:

(1) 

$|\overrightarrow{H}{|}^2=2$ and $M^n$ is locally $S^n\left({\displaystyle \frac{1}{\sqrt{3}}}\right)$, or

(2) 

$|\overrightarrow{H}{|}^2\in (0,t_0]$, and $|\overrightarrow{H}{|}^2=t_0$ holds if and only if M is locally $S^{n-1}\left(r\right)\times S^1\left(\sqrt{1-r^2}\right)$, where r is given by (48) and $t_0$ is the unique real root in $(0,2)$ of the polynomial

$$f_{n,3}\left(t\right)=n^4{t}^3-2n^2(n^2-5n+5)t^2+(1-n)(2n-5)(3n-5)t+(1-n){(n-2)}^2.$$

In the same paper, Chen and Guan also obtained the following two Corollaries.

Corollary 11

([117]). If $M^n$ is a proper CMC triharmonic hypersurface of $S^{n+1}(n\ge 3)$ with two distinct principal curvatures, then M is locally $S^k\left(r\right)\times S^{n-k}\left(\sqrt{1-r^2}\right)$, where $r^2$ is the unique real root of the polynomial:

$$P_{n,k}\left(x\right)=3n{x}^3-(2n+5k)x^2+5kx-k.$$

(49)

Corollary 12

([117]). If $M^n$ is a closed CMC triharmonic hypersurface of $S^{n+1}(n\ge 3)$ with exactly three distinct principal curvatures, then M is minimal.

Chen and Guan also achieved the following complete classification of complete, proper, CMC, triharmonic hypersurfaces in $S^4$.

Theorem 100

([117]). If $M^3$ is a complete proper CMC triharmonic hypersurface in $S^4$, then $M^3$ is one of the following:

(1) 

$M^3=S^3\left(\frac{1}{\sqrt{3}}\right)$;

(2) 

$M^3=S^2\left(r\right)\times S^1\left(\sqrt{1-r^2}\right)$, where $r^2\approx 0.389833$ is the unique real root of $f\left(x\right)=9x^3-16x^2+10x-2$.

In [118], Chen and Guan improved Theorems 98 and 99 to CMC, proper triharmonic hypersurfaces in a real space form ($R^{n+1}\left(c\right)$) with, at most, four distinct principal curvatures. They also obtained the following.

Theorem 101

([118]). Let $M^n$ be a CMC, proper triharmonic hypersurface in $R^{n+1}\left(c\right)(n\ge 4)$. If zero is a principal curvature of $M^n$ with multiplicity of, at most, one, then $M^n$ has constant scalar curvature.

Theorem 102

([118]). If $M^4$ is a CMC, proper, triharmonic hypersurface in $R^5\left(c\right)$, then $M^4$ has constant scalar curvature.

Montaldo, Oniciuc, and Ratto proved that a CMC, proper, triharmonic surface in $S^3$ is locally the totally umbilical sphere ($S^2\left(\frac{1}{\sqrt{3}}\right)$) (see [119], Theorem 1.6). In the same paper, they proved the following.

Theorem 103

([119], Theorem 1.9). Let $M^n$ be a CMC, proper k-harmonic hypersurface in $S^{n+1}(n\ge 3)$ with $k\ge 3$ and constant scalar curvature. Then, we have $|\overrightarrow{H}{|}^2\in (0,k-1]$.

Moreover, $|\overrightarrow{H}{|}^2=k-1$ holds if and only if $M^n$ is locally the totally umbilical hypersurface $\left(S^n\left(\frac{1}{\sqrt{k}}\right)\right)$.

21.2. Triharmonic Hypersurfaces in Space Forms Without Restrictions on Principal Curvatures

Y. Fu and D. Yang [120] were able to prove the following results on CMC triharmonic hypersurfaces in real space forms without imposing restrictions on the number of principal curvatures.

Theorem 104

([120]). If $M^n$ is a CMC triharmonic hypersurface in $H^{n+1}$, then it is minimal.

Theorem 105

([120]). If $M^n$ is a CMC proper triharmonic hypersurface in $S^{n+1}$, then it has constant scalar curvature.

For $c=0$, Fu and Yang [120] proved the following.

Theorem 106

([120]). If $M^n$ is a CMC, proper, triharmonic hypersurface of ${\mathbb{E}}^{n+1}$ such that zero is a principal curvature with multiplicity of, at most, one, then it is minimal.

In particular, they obtained the following.

Theorem 107

([120]). Every CMC triharmonic hypersurface of ${\mathbb{E}}^6$ is minimal.

21.3. Stability of Triharmonic Hypersurfaces in Space Forms

In [121], V. Branding derived the second variational formula for triharmonic maps. He also studied the normal stability of triharmonic hypersurfaces in a real space form. In particular, he obtained the following results.

Theorem 108

([121]). Every minimal triharmonic hypersurface of a real space form of constant curvature is always weakly stable for normal variations.

Theorem 109

([121]). Every triharmonic hypersurface of constant mean curvature in a Euclidean space is weakly normally stable, and every triharmonic hypersurface in a hyperbolic space with constant mean curvature is always normally stable.

Theorem 110

([121]). The proper triharmonic hypersphere ($\varphi :S^n\left(\frac{1}{\sqrt{3}}\right)\to S^{n+1}$) has a normal index equal to one.

21.4. Triharmonic Submanifolds of Riemannian Manifolds of Non-Positive Constant Curvature

Let $\varphi :(M,g)\to (N,\tilde{g})$ be a smooth map. $E_4\left(\varphi \right)$ denotes the 4-energy of $\varphi$ (see Section 4). Then, the extended 4-energy (${\tilde{E}}_4\left(\varphi \right)$) is defined by

$$\begin{array}{c}\hfill {\tilde{E}}_4\left(\varphi \right)=2E_4\left(\varphi \right)-{\int }_M{\left|dd\tau \left(\varphi \right)\right|}^2\mathrm{d}{v}_g.\end{array}$$

(50)

In [122], Maeta, Nakauchi, and Urakawa studied the generalized Chen conjecture for triharmonic submanifolds of a real space form of non-positive constant curvature. They proved that if the domain is complete and the 4-energy of $\varphi$ and the $L^4$ norm of the tension field ($\tau \left(\varphi \right)$) are both finite, then such submanifolds must be minimal. More precisely, they obtained the following.

Theorem 111

([122]). If $\varphi :(M^n,g)\to R^m\left(c\right)$ is an isometric immersion of a complete Riemannian n-manifold ($(M^n,g)$) into a Riemannian m manifold ($R^m\left(c\right)$) of non-positively constant curvature c, then we have the following:

(1)

For $c<0$, if $M^n$ is triharmonic and the extended 4-energy (${\tilde{E}}_4\left(\phi \right)=\frac{1}{2}{\int }_M{\left|\overline{\Delta }\tau \left(\varphi \right)\right|}^2\mathrm{d}{v}_g$) and the $L^4$ norm (${\int }_M{\left|\tau \left(\phi \right)\right|}^4\mathrm{d}{v}_g$) are both finite, then ϕ is minimal.

(2)

For $c=0$, the same conclusion holds if we also assume that $E_2\left(\phi \right)=\frac{1}{2}{\int }_M{\left|\tau \left(\phi \right)\right|}^2\mathrm{d}{v}_g<\infty$ or $E_3\left(\phi \right)={\displaystyle \frac{1}{2}}{\int }_M{\left|\overline{\nabla }\tau \left(\phi \right)\right|}^2\mathrm{d}{v}_g<\infty$.

21.5. Triharmonic CMC Hypersurfaces in Pseudo-Riemannian Space Forms

When the ambient manifolds are pseudo-Riemannian, results on triharmonic hypersurfaces change drastically. For instance, Branding et al. [123] provided such examples for non-minimal triharmonic surfaces lying in an anti-de Sitter 3-space. Branding et al. also proved in the same paper that CMC, triharmonic surfaces ($M_r^2$) with a diagonalizable shape operator in ${\mathbb{E}}_s^3$, $S_{r+1}^3$, or $H_r^3$ are always minimal. The same conclusions are also true for hypersurfaces in ${\mathbb{E}}_s^6$, $S_{r+1}^6$, or $H_r^6$ (see [124] for details).

V. Branding et al. also proved the following.

Theorem 112

([123]). For $k\ge 3$ and $n\ge 2$, let $M^n$ be a space-like, CMC, k-harmonic surface in $R_1^{n+1}\left(c\right)(c\ge 0)$. If $\mathrm{Trace}\left(A^2\right)$ is a constant, then $M^n$ is minimal.

Theorem 113

([123]). For $k\ge 3,n\ge 2$ and $1\le t\le n$, let $M_s^n$ be a pseudo-Riemannian, CMC, k-harmonic hypersurface in $R_t^{n+1}\left(c\right)$. If $\mathrm{Trace}\left(A^2\right)$ is a positive constant and $ϵc<0$, then $M_s^n$ is minimal, where ϵ is the inner product of the unit normal vector of $M_s^n$.

Theorem 114

([123]). For $k\ge 3,n\ge 2$ and $1\le t\le n$, let $S_t^n\left(c\right)$ be a a small pseudo-hypersphere in $S_t^{n+1}$. Then, $S_t^n\left(c\right)$ is a proper k-harmonic if and only if we have $c=k$.

Li Du [124] proved the following.

Theorem 115

([124]). If $M_r^5$ is a non-minimal, CMC, triharmonic hypersurface in a 6D pseudo-Riemannian space form with a diagonalizable shape operator, then $M_r^n$ has constant scalar curvature.

Recently, Li Du also investigated CMC triharmonic hypersurfaces in pseudo-Riemannian space forms of higher dimensions. He established the following.

Theorem 116

([125]). Let $M_r^n$ be a non-minimal, CMC, triharmonic hypersurface of a pseudo-Riemannian space form with a diagonalizable shape operator. If $M_r^n$ has, at most, four distinct principal curvatures and the multiplicity of the zero principal curvature is, at most, one, then it has constant scalar curvature.

Theorem 117

([125]). Let $M_r^n$ be a CMC, triharmonic hypersurface in pseudo-Euclidean space with a diagonalizable shape operator. Assume that $M_r^n$ has, at most, four distinct principal curvatures and the multiplicity of the zero principal curvature is, at most, one; then, it is minimal.

In addition, Du [125] established a classification result for non-minimal, CMC, triharmonic hypersurfaces in a pseudo-Riemannian space form with a diagonalizable shape operator and, at most, two distinct principal curvatures. Sun, Yang, and Zhu [126] also investigated triharmonic hypersurfaces ($M_s^n$) in a pseudo-Riemannian space form ($R_t^{n+1}\left(c\right)$). They established the following.

Theorem 118

([126]). Let $M_s^n$ be a non-minimal, CMC, triharmonic hypersurface in a pseudo-Riemannian space form ($R_t^{n+1}\left(c\right)(c\ne 0)$). If the shape operator of $M_s^n$ is diagonalizable, then $M_s^n$ has constant scalar curvature.

Theorem 119

([126]). Let $M_t^n$ be a CMC triharmonic hypersurface of a pseudo-Riemannian space form ($R_s^{n+1}\left(c\right)(0\le s\le n+1)$) with a diagonalizable shape operator. If $ϵc<0$, then $M_t^n$ is minimal, where ϵ is the inner product of the unit normal vector of $M_t^n$.

In 2024, Du and Luo [127] investigated the minimality of triharmonic CMC hypersurfaces in pseudo-Riemannian space forms under the assumption that the shape operator is diagonalizable. They showed that such non-minimal hypersurfaces must have constant scalar curvature.

Theorem 120

([127]). Let $M_r^n$ be a non-minimal CMC triharmonic hypersurface in ${\mathbb{E}}_t^{n+1}$ such that the multiplicity of the zero principal curvature is, at most, one. Then, $M_r^n$ must be minimal.

Theorem 121

([127]). If $M_r^n(n>5)$ is a CMC triharmonic hypersurface in $S_{r+1}^{n+1}\left(c\right)$ or $H_r^{n+1}\left(c\right)$ with a diagonalizable shape operator, then $M_r^n$ must be minimal.

Theorem 122

([127]). If $M_r^n(n>5)$ is a non-minimal, CMC triharmonic hypersurface in $R_s^{n+1}\left(c\right)(c\ne 0)$ with a diagonalizable shape operator, then $M_r^n$ has constant scalar curvature (τ). Furthermore,

(a)

If $c>0$, then the mean curvature vector of $M_r^n$ is space-like, and τ satisfies

$$n^{2}c-n^2|\overrightarrow{H}{|}^2<\tau \le n(n+1)c-n^2{|\overrightarrow{H}|}^2$$

with $|\overrightarrow{H}{|}^2\le 2c$. Both equalities hold if and only if $M_r^n$ is an open portion of $S_r^n\left(3c\right)$;

(b)

If $c<0$, then the mean curvature vector is time-like, and τ satisfies

$$n(n+1)c+n^2|\overrightarrow{H}{|}^2\le \tau <n^{2}c+n^2{|\overrightarrow{H}|}^2$$

with $|\overrightarrow{H}{|}^2\le -2c$. Both equalities hold if and only if $M_r^n$ is an open portion of $H_r^n\left(3c\right)$.

Remark 11.

The corresponding results of Theorem 122 for $n=5$ were previously obtained by Du [124].