Classification and Existence of f-Biharmonic Hypersurfaces in Nonpositively Curved Conformally Flat Spaces

Abstract

We first study the f-biharmonicity of totally umbilical hypersurfaces in a Riemannian manifold of dimension $n\ge 3$ and prove that any totally umbilical proper f-biharmonic hypersurface without boundaries in a nonpositively curved manifold must be noncompact. Since biharmonic submanifolds are special cases of f-biharmonic submanifolds, our results on f-biharmonic hypersurfaces in nonpositively curved conformally flat spaces provide a natural extension of the generalized Chen’s conjecture. We then investigate the f-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Next, we study f-biharmonic surfaces in a conformally flat 3-space, and for those with nonzero constant mean curvature (CMC), we provide a complete classification of them in 3-space forms. Finally, we investigate the f-biharmonicity of hypersurfaces in a conformally flat space with negative sectional curvature. Our results generalize some previous conclusions on biharmonic hypersurfaces.

1. Introduction and Preliminaries

All maps, manifolds, and tensor fields studied in this paper are assumed to be smooth unless stated otherwise.

A biharmonic map is one $\phi :(M^m,g)\to (N^n,h)$ from an m-dimensional to an n-dimensional Riemannian manifold whose bitension field vanishes identically (see [1]), i.e.,

$${\tau }_2\left(\phi \right):={\mathrm{Trace}}_g({\nabla }^{\phi }{\nabla }^{\phi }-{\nabla }_{{\nabla }^M}^{\phi })\tau \left(\phi \right)-{\mathrm{Trace}}_g{R}^N(\mathrm{d}\phi ,\tau \left(\phi \right))\mathrm{d}\phi \equiv 0.$$

(1)

Here, ${\mathrm{Trace}}_g$ denotes the trace operation with respect to the Riemannian metric g and $\tau \left(\phi \right):={\mathrm{Trace}}_g\nabla \mathrm{d}\phi$. Moreover, ${\mathrm{R}}^N$ denotes the curvature operator of $(N,h)$ given by

$${\mathrm{R}}^N(X,Y)Z=[{\nabla }_X^N,{\nabla }_Y^N]Z-{\nabla }_{[X,Y]}^{N}Z.$$

An f-biharmonic map is one whose f-bitension field satisfies the PDE system (see, e.g., [2,3])

$${\tau }_{2,f}\left(\phi \right):=f{\tau }_2\left(\phi \right)+(\Delta f)\tau \left(\phi \right)+2{\nabla }_{\mathrm{grad}f}^{\phi }\tau \left(\phi \right)=0,$$

(2)

where ${\tau }_2\left(\phi \right)$ is the bitension field of $\phi$ and $f:M\to (0,+\infty )$.

From the variational point of view, it can be seen that a biharmonic map is a critical point of the bienergy functional $E_2\left(\phi \right)={\displaystyle \frac{1}{2}}{\int }_M{\left|\tau \left(\phi \right)\right|}^2\mathrm{d}{v}_g$ whilst an f-biharmonic map is a critical point of the f-bienergy functional $E_{2,f}\left(\phi \right)={\displaystyle \frac{1}{2}}{\int }_{M}f{\left|\tau \left(\phi \right)\right|}^2\mathrm{d}{v}_g$.

Here, f acts as a weight function that generalizes the classical biharmonic equation by introducing a variable coefficient, which allows us to investigate a larger class of maps and submanifolds while retaining a natural variational structure. As weighted operators arise frequently in conformal geometry, geometric analysis, and submanifold theory under curvature restrictions, the f-biharmonic condition provides a natural and flexible extension of biharmonicity.

It is well known that a map $\phi$ between Riemannian manifolds is called a harmonic map if and only if its tension field vanishes identically (see, e.g., [4]). Clearly, any harmonic map is a biharmonic map, whilst any biharmonic map is an f-biharmonic map. Thus, we see that f-biharmonic maps generalize the concepts of harmonic maps and biharmonic maps. The following strict inclusion relations hold among them:

$$\left\{Harmonicmaps\right\}\subset \left\{Biharmonicmaps\right\}\subset \{f-biharmonicmaps\}.$$

We call non-biharmonic f-biharmonic maps proper f-biharmonic maps.

Recall that a submanifold is called a biharmonic (resp., an f-biharmonic) submanifold if the isometric immersion defining it is a biharmonic map (resp., an f-biharmonic map). Clearly, every biharmonic submanifold is an f-biharmonic submanifold, but an f-biharmonic submanifold with a nonconstant positive function f is not biharmonic. From [5], the biharmonic equation for a hypersurface in a generic Riemannian manifold $(N^{m+1},h)$ (defined by an isometric immersion $\phi :(M^m,g)\to (N^{m+1},h)$) can be stated as follows:

Theorem 1 ([5]).

Let $\varphi :M^m\to N^{m+1}$ be a hypersurface with mean curvature vector field $\eta =H\xi$. Then, ϕ is biharmonic if and only if

$$\left\{\begin{array}{c}\Delta H-{H\left|A\right|}^2+H{\mathrm{Ric}}^N(\xi ,\xi )=0,\hfill \\ 2A\left(\mathrm{grad}H\right)+{\displaystyle \frac{m}{2}}\mathrm{grad}{H}^2-2H{\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }=0.\hfill \end{array}\right.$$

Here, A denotes the shape operator of the hypersurface with respect to the unit normal vector field ξ and ${\mathrm{Ric}}^N:T_{q}N⟶T_{q}N$ is the Ricci operator of the ambient space defined by $⟨{\mathrm{Ric}}^N\left(Z\right),W⟩={\mathrm{Ric}}^N(Z,W)$.

If we adopt the same notations and sign convention as in Theorem 1, the f-biharmonic equation for hypersurfaces in a generic Riemannian manifold $(N^{m+1},h)$ can be stated as follows:

Theorem 2 ([3,6]).

A hypersurface $\varphi :(M^m,g)\to (N^{m+1},h)$ with mean curvature vector field $\eta =H\xi$ is f-biharmonic if and only if

$$\left\{\begin{array}{c}\Delta \left(fH\right)-\left(fH\right)\left[\right|A{|}^2-{\mathrm{Ric}}^N(\xi ,\xi )]=0,\hfill \\ A\left(\mathrm{grad}\left(fH\right)\right)+\left(fH\right)[{\displaystyle \frac{m}{2}}\mathrm{grad}H-{\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }]=0,\hfill \end{array}\right.$$

(3)

which is equivalent to

$$\left\{\begin{array}{c}\Delta H-{H\left|A\right|}^2+H{\mathrm{Ric}}^N(\xi ,\xi )+H(\Delta f)/f+2\left(\mathrm{grad}\ln f\right)H=0,\hfill \\ 2A\left(\mathrm{grad}H\right)+{\displaystyle \frac{m}{2}}\mathrm{grad}{H}^2-2H{\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }+2HA\left(\mathrm{grad}\ln f\right)=0.\hfill \end{array}\right.$$

We previously remarked that f-biharmonic maps are generalizations of biharmonic maps. Naturally, f-biharmonic submanifolds generalize the notion of biharmonic submanifolds.

The study of biharmonic submanifolds focuses on a fundamental problem: classifying biharmonic submanifolds in certain model spaces, such as space forms, conformally flat spaces and homogeneous spaces. Though significant progress has been made, the following conjecture remains open in general cases.

Chen’s Conjecture ([7]): Any biharmonic submanifold of Euclidean space ${\mathbb{R}}^n$ is harmonic (i.e., minimal).

We remark that the ambient space in the original conjecture is Euclidean space, which provides a contrast to the curved spaces considered in this paper. Another interesting problem in the study of biharmonic submanifolds is the following conjecture.

The Generalized Chen’s Conjecture ([8]): Any biharmonic submanifold in $(N^n,h)$ with nonpositive sectional curvature is minimal.

For more examples on biharmonic submanifolds, as well as basic examples and properties of biharmonic maps, we refer the reader to [2,3,6,9,10,11,12,13,14,15,16,17,18,19] and the references therein.

We would like to point out that the generalized Chen’s conjecture has been proved to be false in [16] by constructing proper biharmonic hyperplanes in a five-dimensional conformally flat space of negative sectional curvature. In that paper, the authors used this result and the biharmonicity of the product maps to construct infinitely many examples of proper biharmonic submanifolds in nonpositively curved Riemannian manifolds. As a generalization of biharmonic submanifolds, following the idea in [16], we focus on studying f-biharmonic submanifolds in conformally flat spaces, where a suitable conformal factor can be chosen to ensure their existence. We then investigate whether there exist proper f-biharmonic submanifolds in nonpositively curved Riemannian manifolds.

On the other hand, it is a fact that there is an interesting relationship among f-biharmonicity, biharmonicity and conformality from a two-dimensional Riemannian manifold to a Riemannian manifold (see [3], Theorem 2.3 and Corollary 3.6, for details). Given this relationship, it would be interesting to study f-biharmonic submanifolds in certain model spaces. For some progress on f-biharmonic submanifolds, as well as basic examples and properties of f-biharmonic maps, we refer the readers to [2,3,6,15,20,21,22,23,24] and the book [17] and the references therein.

We call a Riemannian manifold $(N^n,h)$ a conformally flat space if for any point of $N^n$ there is a neighborhood that is conformally diffeomorphic to an open subset of the Euclidean space ${\mathbb{R}}^n$. Note that all two-dimensional Riemannian manifolds, space forms ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$, are conformally flat spaces.

In this paper, we will focus on the classification and existence of f-biharmonic hypersurfaces in nonpositively curved conformally flat spaces.

This paper is organized as follows:

In Section 2, we first study totally umbilical f-biharmonic hypersurfaces in Riemannian manifolds of dimension $n\ge 3$, and then we investigate the f-biharmonicity of totally umbilical hypersurfaces in conformally flat spaces. In Section 3, we employ f-biharmonic surfaces to construct biharmonic conformal immersions, and we conversely produce f-biharmonic surfaces from prescribed conformal immersions. Meanwhile, we establish a complete classification of f-biharmonic surfaces with nonzero constant mean curvature in three-dimensional space forms. Section 4 focuses on the construction of explicit examples of proper f-biharmonic submanifolds in nonpositively curved manifolds. Following the research ideas in [12,16], we construct totally umbilical proper f-biharmonic hypersurfaces $M^m$ ($m\ge 3,m\ne 4$) in conformally flat spaces with strictly negative sectional curvature, which provides concrete examples of proper f-biharmonic submanifolds in nonpositively curved manifolds. Finally, Section 5 concludes the paper, summarizes the main geometric results, and presents some prospective research problems for future study.

2. Totally Umbilical $\mathit{f}$-Biharmonic Hypersurfaces

In this section, we first study totally umbilical f-biharmonic hypersurfaces in a Riemannian manifold. Then, we investigate the f-biharmonicity of a family of totally umbilical hypersurfaces in conformally flat spaces. These hypersurfaces are constructed by starting with a totally geodesic hyperplane in a Euclidean space and then performing a suitable conformal change of the Euclidean metric into a conformally flat metric. More specifically, following [12,16], we analyze the f-biharmonicity of a totally umbilical hyperplane $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h)$ into a conformally flat space with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$, where $a_i$ are constants.

We are now ready to provide a characterization of totally umbilical f-biharmonic hypersurfaces in a general Riemannian manifold.

Theorem 3.

A totally umbilical hypersurface $\phi :(M^m,g)\to (N^{m+1},h)$ with mean curvature vector field $\eta =H\xi$ is f-biharmonic if and only if one of the following cases holds:

(i) The hypersurface is totally geodesic;

(ii) $m=4$ or $H=\mathit{constant}\ne 0$, in which case the hypersurface is biharmonic. In particular, if $H=\mathit{constant}\ne 0$, then ${\mathrm{Ric}}^N(\xi ,\xi )={\left|A\right|}^2=m{H}^2>0$;

(iii) $m\ne 4$, $f={c\left|H\right|}^{{\displaystyle \frac{m-4}{2}}}$, and H is nonconstant satisfying the following PDE:

$${\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }=(m-1)\mathrm{grad}H,{\mathrm{Ric}}^N(\xi ,\xi )=m{H}^2-{\left|H\right|}^{{\displaystyle \frac{2-m}{2}}}{\Delta \left(\right|H|}^{{\displaystyle \frac{m-2}{2}}})$$

(4)

on the hypersurface, where c is a positive constant. Moreover, by the totally umbilical property of the hypersurface, the first equation of (4) holds naturally, and the hypersurface is proper f-biharmonic in this case.

Proof. 

An orthonormal frame $\{e_1,\dots ,e_m,\xi \}$ of an $(m+1)$-dimensional Riemannian manifold $N^{m+1}$ is adapted to the totally umbilical hypersurface $M^m$ with $\eta =H\xi$ as the mean curvature vector field. It follows that $A\left(e_i\right)=H{e}_i$, $i=1,2,\dots m$, ${\left|A\right|}^2=m{H}^2$ and $A\left(\mathrm{grad}H\right)=H\mathrm{grad}H.$

A straightforward computation yields the Ricci curvatures:

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m}⟨\mathrm{R}(e_i,e_j)e_i,\xi ⟩={\displaystyle \sum _{i=1}^m}\mathrm{R}(\xi ,e_i,e_i,e_j)=-{\mathrm{Ric}}^N(e_j,\xi ),j=1,2,\dots ,m.\end{array}$$

(5)

Since $\left\{e_i\right\}$ are principal directions with principal curvature H, one can verify that

$$\begin{array}{ccc}\hfill \left({\nabla }_{e_i}B\right)(e_j,e_i)& =& e_i\left(B(e_j,e_i)\right)-B({\nabla }_{e_i}{e}_j,e_i)-B({\nabla }_{e_i}{e}_i,e_j)\hfill \\ \hfill & =& -B(e_i,e_i)⟨{\nabla }_{e_i}{e}_j,e_i⟩-B(e_j,e_j)⟨{\nabla }_{e_i}{e}_i,e_j⟩\hfill \\ \hfill & =& -H(⟨{\nabla }_{e_i}{e}_j,e_i⟩+⟨{\nabla }_{e_i}{e}_i,e_j⟩)=0,\hfill \end{array}$$

(6)

and

$$\begin{array}{c}\left({\nabla }_{e_j}B\right)(e_i,e_i)=e_j\left(B(e_i,e_i)\right)-B({\nabla }_{e_j}{e}_i,e_i)-B({\nabla }_{e_j}{e}_i,e_i)\hfill \\ =e_j\left(H\right)-2B(e_i,e_i)⟨{\nabla }_{e_j}{e}_i,e_i⟩=e_j\left(H\right),forj=1,2,\dots m,\hfill \end{array}$$

(7)

where B denotes the second fundamental form of the hypersurface and the covariant derivative of B is defined by

$$(\nabla B)(X,Y,Z)=\left({\nabla }_{X}B\right)(Y,Z)=X\left(B(Y,Z)\right)-B({\nabla }_{X}Y,Z)-B({\nabla }_{X}Z,Y).$$

Subtracting (7) from (6) and applying (5) together with the Codazzi equation of the hypersurface,

$$\left({\nabla }_{X}B\right)(Y,Z)-\left({\nabla }_{Y}B\right)(X,Z)={\left({\mathrm{R}}^N(X,Y)Z\right)}^{\perp }=⟨{\mathrm{R}}^N(X,Y)Z,\xi ⟩,$$

we obtain

$$\begin{array}{c}\hfill e_j\left(H\right)={\mathrm{R}}^N(\xi ,e_i,e_j,e_i),j=1,2,\dots ,m.\end{array}$$

(8)

This implies $(m-1)e_j\left(H\right)={\mathrm{Ric}}^N(e_j,\xi ),j=1,2,\dots m,$ and consequently

$$\begin{array}{c}(m-1)\mathrm{grad}H={\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }.\hfill \end{array}$$

(9)

It follows that the first equation of (4) holds naturally for totally umbilical hypersurfaces.

Clearly, $H=0$ implies that the hypersurface is totally geodesic.

Hereafter, we assume $H\ne 0$.

Substituting (9) into the 2nd equation of (3) and simplifying the resulting expression gives

$${\mathrm{grad}\left(\ln \right(f\left|H\right|}^{{\displaystyle \frac{4-m}{2}}}\left)\right)={0,i.e.,f\left|H\right|}^{{\displaystyle \frac{4-m}{2}}}=c$$

(10)

on the hypersurface, where $c>0$ is a constant. This shows that either $m=4$ or $H=\mathrm{constant}\ne 0$ forces f to be a constant. In this case, Theorem 2 implies the hypersurface is f-biharmonic if and only if it is biharmonic. Moreover, for nonzero constant H, (3) yields ${\mathrm{Ric}}^N(\xi ,\xi )={\left|A\right|}^2=m{H}^2>0$.

Now, suppose that $m\ne 4$ and H is nonconstant. Substituting (10) into the first equation of (3) and simplifying the result leads to

$${\mathrm{Ric}}^N(\xi ,\xi )=m{H}^2-{\left|H\right|}^{{\displaystyle \frac{2-m}{2}}}{\Delta \left(\right|H|}^{{\displaystyle \frac{m-2}{2}}}).$$

(11)

Summarizing the above arguments, the theorem is established. □

Remark 1.

In the proof of Theorem 3, combining the f-biharmonic condition, the umbilical property, nonvanishing mean curvature and the Codazzi equation, we derive Equation (10), which yields

$${f\left|H\right|}^{{\displaystyle \frac{4-m}{2}}}=c$$

on the hypersurface, where c is a positive constant. This implies that for $m=4$, the exponent ${\displaystyle \frac{4-m}{2}}$ vanishes, which immediately gives $f=c$. Consequently, f is necessarily constant regardless of H, the coupling between f and H disappears, and the f-biharmonic condition imposes no additional restriction on H beyond the umbilical hypothesis.

As immediate consequences of Theorem 3, we obtain the following corollaries:

Corollary 1.

A totally umbilical hypersurface $\phi :M^m\to (N^{m+1},h)$ with mean curvature vector field $\eta =H\xi$ is proper f-biharmonic if and only if $m\ne 4$, $f={c\left|H\right|}^{{\displaystyle \frac{m-4}{2}}}$ and H is nonconstant, satisfying

$${\mathrm{Ric}}^N(\xi ,\xi )=m{H}^2-{\left|H\right|}^{{\displaystyle \frac{2-m}{2}}}{\Delta \left(\right|H|}^{{\displaystyle \frac{m-2}{2}}}),{\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }=(m-1)\mathrm{grad}H$$

(12)

on the hypersurface, where $c>0$ is a constant.

Corollary 2

([6]). Any totally umbilical f-biharmonic hypersurface in a hyperbolic space ${\mathbb{H}}^{m+1}$ or a Euclidean space ${\mathbb{R}}^{m+1}$ is totally geodesic; a totally umbilical f-biharmonic hypersurface in a sphere ${\mathbb{S}}^{m+1}$ is a part of the small sphere ${\mathbb{S}}^m({\displaystyle \frac{1}{\sqrt{2}}})$ or a part of the great sphere ${\mathbb{S}}^m\subset {\mathbb{S}}^{m+1}$.

Proof. 

Let $M^m\to N^{m+1}\left(C\right)$ be a totally umbilical f-biharmonic hypersurface with mean curvature vector field $\eta =H\xi$ in a space form with constant sectional curvature $C=-1,0,1$. Hence, ${\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }=0$. By Theorem 3, the hypersurface is a totally umbilical biharmonic hypersurface. It is well known that such biharmonic hypersurfaces in hyperbolic spaces and Euclidean spaces are totally geodesic (see, e.g., [1,7]), and a totally umbilical biharmonic hypersurface in ${\mathbb{S}}^{m+1}$ is a part of the small sphere ${\mathbb{S}}^m({\displaystyle \frac{1}{\sqrt{2}}})$ in ${\mathbb{S}}^{m+1}$ or a part of the great sphere ${\mathbb{S}}^m\subset {\mathbb{S}}^{m+1}$ ([8,10]). □

Remark 2.

(i) Corollary 2 was also proved in [6] in a different way.

(ii) By Theorem 3 and Corollary 1, it follows that any totally umbilical four-dimensional hypersurface in $N^5$ cannot be proper f-biharmonic, and any totally umbilical proper f-biharmonic hypersurface has nonconstant mean curvature. Note that a totally umbilical biharmonic hypersurface $M^m$ in $N^{m+1}$ with $m\ne 4$ must have constant mean curvature ([10,25]).

It is interesting to compare Lemma 3.1 in [16] and Theorem 3.4 in [12], which provided many examples of totally umbilical proper biharmonic four-dimensional hypersurfaces with nonconstant mean curvature in conformally flat 5-spaces.

For $m=2$, our Theorem 3 recovers Theorem 2.1 in [23].

Applying Corollary 1, we obtain the following proposition:

Proposition 1.

Any totally umbilical proper f-biharmonic hypersurface without boundary in a nonpositively curved manifold must be a noncompact m-dimensional manifold with $m\ge 3$ and $m\ne 4$.

Proof. 

Let $\phi :(M^m,g={\phi }^{\ast }h)\to (N^{m+1},h)$ be a totally umbilical hypersurface with mean curvature vector field $\eta =H\xi$ in a nonpositively curved manifold. This implies ${\mathrm{Ric}}^N(\xi ,\xi )\le 0$.

We show that if the hypersurface is proper f-biharmonic, then it must be noncompact. In fact, by Corollary 1, proper f-biharmonicity implies $\left|H\right|>0$ (nonconstant) and $m\ne 4$. Combining these conditions, a straightforward computation using (12) yields

$${\Delta \left(\right|H|}^{{\displaystyle \frac{m-2}{2}}}{)=m|H|}^{{\displaystyle \frac{m+2}{2}}}-{\left|H\right|}^{{\displaystyle \frac{m-2}{2}}}{\mathrm{Ric}}^N(\xi ,\xi )\ge m{\left|H\right|}^{{\displaystyle \frac{m+2}{2}}}>0$$

(13)

on the hypersurface.

For $m=2$, substituting into (13) gives $0=\Delta ({\left|H\right|}^{{\displaystyle \frac{m-2}{2}}}){\ge m\left|H\right|}^{{\displaystyle \frac{m+2}{2}}}>0$, which is a contradiction. For $m\ge 3$, (13) implies that ${\left|H\right|}^{{\displaystyle \frac{m-2}{2}}}$ is a strictly subharmonic function on the hypersurface. If the hypersurface without boundaries is compact, the Hopf theorem would force ${\left|H\right|}^{{\displaystyle \frac{m-2}{2}}}$ to be constant; hence, H is constant, which contradicts $\left|H\right|>0$ (nonconstant). Thus, the proposition is established. □

Remark 3.

For $m\ge 3$ and $m\ne 4$, consider a totally umbilical proper f-biharmonic m-dimensional hypersurface without boundaries in a nonpositively curved manifold. In the proof of Theorem 3, the inequality ${\Delta \left(\right|H|}^{{\displaystyle \frac{m-2}{2}}}{)\ge m|H|}^{{\displaystyle \frac{m+2}{2}}}>0$ holds on the hypersurface, where H is the mean curvature. This inequality forces ${\left|H\right|}^{{\displaystyle \frac{m-2}{2}}}$ to grow at least polynomially if it is unbounded.

We need the following technical proposition.

Proposition 2.

For constants $c>0$, $a_i$$(i=1,2,\dots ,m+1)$ and $m\ne 4$, let $h_0={\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2$. A hypersurface $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h={\beta }^{-2}{h}_0)$ defined by $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is proper f-biharmonic if and only if the function $f=c{(1+{\displaystyle \sum _{j=1}^m}{a}_j^2)}^{{\displaystyle \frac{4-m}{4}}}{|-{\displaystyle \sum _{i=1}^m}{a}_i{\beta }_i+{\beta }_z|}^{{\displaystyle \frac{m-4}{2}}}$ is nonconstant, and β satisfies the following PDE:

$$\begin{array}{c}{\displaystyle \sum _{i=1}^m}(\beta {\beta }_{ii}-m{\beta }_i^2)+\beta {\beta }_{zz}-m{\beta }_z^2\hfill \\ +(m-1)({\displaystyle \sum _{i,j=1}^m}{a}_i{a}_j\beta {\beta }_{ij}-2{\displaystyle \sum _{i=1}^m}{a}_i\beta {\beta }_{iz}+\beta {\beta }_{zz})/(1+{\displaystyle \sum _{i=1}^m}{a}_i^2)\hfill \\ =m{H}^2-{\displaystyle \frac{m-2}{2\left|H\right|}}\{{\beta }^2{\Delta }_{h_0}\left|H\right|+(m-2)\beta [H{\xi }_0\left(\right|H\left|\right)-{\mathrm{grad}}_{h_0}\beta \left(\right|H\left|\right)]-{\beta }^2{\xi }_0{\xi }_0\left(\right|H\left|\right)\}\hfill \\ -{\displaystyle \frac{(m-2)(m-4)}{4H^2}}{\beta }^2\left\{\right|{\mathrm{grad}}_{h_0}H{|}^2-{\left[{\xi }_0\left(H\right)\right]}^2\}\hfill \end{array}$$

(14)

on the hypersurface.

Here, ${\Delta }_{h_0}$ is the Laplacian operator with respect to the metric $h_0$, ${\xi }_0={\displaystyle \frac{-{\displaystyle \sum _{i=1}^m}{a}_i\frac{\partial }{\partial x_i}+\frac{\partial }{\partial z}}{\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2}}}$, $H={\xi }_0\left(\beta \right)$, ${\beta }_i={\displaystyle \frac{\partial \beta }{\partial x_i}}$, ${\beta }_{ij}={\displaystyle \frac{{\partial }^2\beta }{\partial x_i\partial x_j}}$, ${\beta }_{iz}={\displaystyle \frac{{\partial }^2\beta }{\partial x_i\partial z}}$ and ${\beta }_{zz}={\displaystyle \frac{{\partial }^2\beta }{\partial z^2}}$.

Proof. 

It is straightforward to verify that the hypersurface $\phi :({\mathbb{R}}^m,g_0={\phi }^{\ast }{h}_0)\to ({\mathbb{R}}^{m+1},h_0={\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2)$ defined by $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is totally geodesic with the unit normal vector field ${\xi }_0={\displaystyle \frac{-{\displaystyle \sum _{i=1}^m}{a}_i\frac{\partial }{\partial x_i}+\frac{\partial }{\partial z}}{\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2}}}$. Consequently, $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h={\beta }^{-2}({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2))$ is a totally umbilical hypersurface with the unit normal vector field $\xi =\beta {\xi }_0$ and mean curvature $H={\xi }_0\left(\beta \right)=\xi \left(\ln \beta \right)={\displaystyle \frac{-{\displaystyle \sum _{i=1}^m}{a}_i{\beta }_i+{\beta }_z}{\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2}}}$ (see, e.g., [13,26]), where ${\beta }_i={\displaystyle \frac{\partial \beta }{\partial x_i}}$.

With respect to the metric h, we apply Claim II in [16] to compute the Ricci curvature:

$$\begin{array}{c}{\mathrm{Ric}}^N(\xi ,\xi )={\Delta }_h\ln \beta +(m-1)[{\mathrm{Hess}}_h\left(\ln \beta \right)(\xi ,\xi )-{\left(\xi \ln \beta \right)}^2+|{\mathrm{grad}}_h\ln \beta {|}_h^2]\hfill \\ =\beta {\Delta }_{h_0}\beta -m{\left|{\mathrm{grad}}_{h_0}\beta \right|}_{h_0}^2+(m-1)\beta {\mathrm{Hess}}_{h_0}\left(\beta \right)({\xi }_0,{\xi }_0)\hfill \\ ={\displaystyle \sum _{i=1}^m}(\beta {\beta }_{ii}-m{\beta }_i^2)+(\beta {\beta }_{zz}-m{\beta }_z^2)+(m-1){\displaystyle \frac{{\displaystyle \sum _{i,j=1}^m}{a}_i{a}_j\beta {\beta }_{ij}-2{\displaystyle \sum _{i=1}^m}{a}_i\beta {\beta }_{iz}+\beta {\beta }_{zz}}{1+{\displaystyle \sum _{i=1}^m}{a}_i^2}},\hfill \end{array}$$

(15)

where ${\Delta }_{h_0}$ and ${\Delta }_h$ denote the Laplacian with respect to the metrics $h_0$ and h, respectively, and ${\beta }_{ij}={\displaystyle \frac{{\partial }^2\beta }{\partial x_i\partial x_j}}$, ${\beta }_{iz}={\displaystyle \frac{{\partial }^2\beta }{\partial x_i\partial z}}$, ${\beta }_{zz}={\displaystyle \frac{{\partial }^2\beta }{\partial z^2}}$.

By taking an orthonormal frame ${\{e_i,\xi \}}_{i=1,2,\dots ,m}$ adapted to the hypersurface $({\mathbb{R}}^m,g={\phi }^{\ast }h)$, a straightforward computation gives

$$\begin{array}{c}{\Delta }_g\left|H\right|=e_i{e}_i\left(\right|H\left|\right)-{\nabla }_{e_i}^M{e}_i\left(\right|H\left|\right)\hfill \\ =e_i{e}_i\left(\right|H\left|\right)+\xi \xi \left(\right|H\left|\right)-{\nabla }_{e_i}^h{e}_i\left(\right|H\left|\right)-{\nabla }_{\xi }^h\xi \left(\right|H\left|\right)\hfill \\ +h(e_i,e_i)\xi \left(\right|H\left|\right)-\xi \xi \left(\right|H\left|\right)+{\nabla }_{\xi }^h\xi \left(\right|H\left|\right)\hfill \\ ={\Delta }_h\left|H\right|+mH\xi \left(\right|H\left|\right)-\xi \xi \left(\right|H\left|\right)+{\nabla }_{\xi }^h\xi \left(\right|H\left|\right)\hfill \\ ={\beta }^2{\Delta }_{h_0}\left|H\right|-\beta (m-1)⟨{\mathrm{grad}}_{h_0}\beta ,{\mathrm{grad}}_{h_0}{\left|H\right|⟩}_{h_0}+mH\beta {\xi }_0\left(\right|H\left|\right)\hfill \\ -\beta {\xi }_0[\beta {\xi }_0\left(\right|H\left|\right)]+{\nabla }_{\xi }^{h_0}\xi \left(\right|H\left|\right)-2\beta {\xi }_0\left(\ln \beta \right)\xi \left(\right|H\left|\right)+{\beta }^2{\mathrm{grad}}_{h_0}\ln \beta \left(\right|H\left|\right)\hfill \\ ={\beta }^2{\Delta }_{h_0}\left|H\right|-(m-2)\beta ⟨{\mathrm{grad}}_{h_0}\beta ,{\mathrm{grad}}_{h_0}{\left|H\right|⟩}_{h_0}\hfill \\ +mH\beta {\xi }_0\left(\right|H\left|\right)-{\beta }^2{\xi }_0{\xi }_0\left(\right|H\left|\right)-2\beta H{\xi }_0\left(\right|H\left|\right),\hfill \end{array}$$

(16)

and

$$\begin{array}{c}|{\mathrm{grad}}_g{\left(\right|H\left|\right)|}^2=⟨{\mathrm{grad}}_h\left|H\right|-\xi \left(\right|H\left|\right)\xi ,{\mathrm{grad}}_h\left|H\right|-\xi \left(\right|H\left|\right)\xi ⟩\hfill \\ =|{\mathrm{grad}}_h{H|}^2+{\left[\xi \left(H\right)\right]}^2-2⟨{\mathrm{grad}}_h\left|H\right|,\xi \left(\right|H\left|\right)\xi ⟩={\beta }^2{\left|{\mathrm{grad}}_{h_0}H\right|}^2-{\beta }^2{\left[{\xi }_0\left(H\right)\right]}^2.\hfill \end{array}$$

(17)

A direct computation thus yields

$$\begin{array}{c}m{H}^2-{\left|H\right|}^{{\displaystyle \frac{2-m}{2}}}{\Delta }_g{\left(\right|H|}^{{\displaystyle \frac{m-2}{2}}})\hfill \\ =m{H}^2-{\displaystyle \frac{m-2}{2}}{\left|H\right|}^{-1}{\Delta }_g\left|H\right|-{\displaystyle \frac{(m-2)(m-4)}{4}}{\left|H\right|}^{-2}|{\mathrm{grad}}_g{\left(\right|H\left|\right)|}^2\hfill \\ =m{H}^2-{\displaystyle \frac{m-2}{2\left|H\right|}}\{{\beta }^2{\Delta }_{h_0}\left|H\right|+(m-2)\beta [H{\xi }_0\left(\right|H\left|\right)-{\mathrm{grad}}_{h_0}\beta \left(\right|H\left|\right)]-{\beta }^2{\xi }_0{\xi }_0\left(\right|H\left|\right)\}\hfill \\ -{\displaystyle \frac{(m-2)(m-4)}{4H^2}}{\beta }^2\left\{\right|{\mathrm{grad}}_{h_0}H{|}^2-{\left[{\xi }_0\left(H\right)\right]}^2\}.\hfill \end{array}$$

(18)

For totally umbilical hypersurfaces, ${\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }=(m-1){\mathrm{grad}}_{\mathrm{g}}H$ holds naturally. By combining Corollary 1 with (15) and (18), the proposition is established. □

If $\beta =\beta \left(z\right)$ depends only on the variable z in Proposition 2, the following theorem provides a main technique to construct f-biharmonic hypersurfaces:

Theorem 4.

The hypersurface $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h={\beta }^{-2}\left(z\right)({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2))$ defined by $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is proper f-biharmonic if and only if the function $f=c{k}_m^{{\displaystyle \frac{m-4}{2}}}{\left|{\beta }^{\prime }\left(z\right)\right|}^{{\displaystyle \frac{m-4}{2}}}$ is nonconstant, and $\beta \left(z\right)$ satisfies the following ODE:

$$\begin{array}{c}m(1+k_m^2){\beta }^{\prime 4}+{\displaystyle \frac{(m^2-2m+2)(1-k_m^2)-2m}{2}}\beta {\beta }^{\prime 2}{\beta }^{\prime \prime }\hfill \\ -{\displaystyle \frac{(m-2)(1-k_m^2)}{2}}{\beta }^2{\beta }^{\prime }{\beta }^{\prime \prime \prime }-{\displaystyle \frac{(m-2)(m-4)(1-k_m^2)}{4}}{\beta }^2{\beta }^{\prime \prime 2}=0\hfill \end{array}$$

(19)

on the hypersurface.

Here, $c>0$, $a_1,a_2,\dots ,a_m$ and $a_{m+1}$ are constants, and $k_m=1/\sqrt{1+{\displaystyle \sum _{i=1}^m}{a}_i^2}$.

Proof. 

By Proposition 2, the hypersurface $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h={\beta }^{-2}\left(z\right)({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2))$ defined by $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is totally umbilical with the unit normal vector field $\xi =\beta {\displaystyle \frac{-{\displaystyle \sum _{i=1}^m}{a}_i\frac{\partial }{\partial x_i}+\frac{\partial }{\partial z}}{\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2}}}$ and mean curvature $H=\xi \left(\ln \beta \right)={\beta }^{\prime }\left(z\right)/\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2}$.

For ${\xi }_0=(-{\displaystyle \sum _{i=1}^m}{a}_i{\displaystyle \frac{\partial }{\partial x_i}}+{\displaystyle \frac{\partial }{\partial z}})/(\sqrt{1+{\displaystyle \sum _{j=1}^m}{a}_j^2})$ and $h_0={\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2$, a straightforward computation gives

$$\begin{array}{c}m{H}^2-{\displaystyle \frac{m-2}{2\left|H\right|}}\{{\beta }^2{\Delta }_{h_0}\left|H\right|+(m-2)\beta [H{\xi }_0\left(\right|H\left|\right)-{\mathrm{grad}}_{h_0}\beta \left(\right|H\left|\right)]-{\beta }^2{\xi }_0{\xi }_0\left(\right|H\left|\right)\}\hfill \\ -{\displaystyle \frac{(m-2)(m-4)}{4H^2}}{\beta }^2\left\{\right|{\mathrm{grad}}_{h_0}H{|}^2-{\left[{\xi }_0\left(H\right)\right]}^2\}\hfill \\ =m{k}_m^2{\beta }^{\prime 2}-{\displaystyle \frac{m-2}{2k_m{\beta }^{\prime }}}[k_m(1-k_m^2){\beta }^2{\beta }^{\prime \prime \prime }+k_m(1-k_m^2)(2-m)\beta {\beta }^{\prime }{\beta }^{\prime \prime }]\hfill \\ -{\displaystyle \frac{(m-2)(m-4)}{4k_m^2{\beta }^{\prime 2}}}\left[k_m^2(1-k_m^2){\beta }^2{\beta }^{\prime \prime 2}\right]\hfill \\ =m{k}_m^2{\beta }^{\prime 2}-{\displaystyle \frac{(m-2)(1-k_m^2)}{2}}{\displaystyle \frac{{\beta }^2{\beta }^{\prime \prime \prime }+(2-m)\beta {\beta }^{\prime }{\beta }^{\prime \prime }}{{\beta }^{\prime }}}-{\displaystyle \frac{(m-2)(m-4)(1-k_m^2)}{4}}{\displaystyle \frac{{\beta }^2{\beta }^{\prime \prime 2}}{{\beta }^{\prime 2}}}.\hfill \end{array}$$

(20)

Note that ${\beta }_i={\displaystyle \frac{\partial \beta }{\partial x_i}}=0,{\beta }_{ij}={\displaystyle \frac{{\partial }^2\beta }{\partial x_i\partial x_j}}=0$ and ${\beta }_{iz}=0$ for $i=1,2,\dots ,m$. Substituting these into (14) along with (20) and simplifying the resulting equation yields (19). Combining this with Proposition 2, we obtain the theorem. □

As an immediate consequence of Theorem 4, we obtain the following corollary:

Corollary 3.

For $\beta =\beta \left(z\right)$, a surface $\phi :({\mathbb{R}}^2,g={\phi }^{\ast }h)\to ({\mathbb{R}}^3,h={\beta }^{-2}\left(z\right)({\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2))$ defined by $\phi (x_1,x_2)=(x_1,x_2,a_1{x}_1+a_2{x}_2+a_3)$ in a conformally flat space is proper f-biharmonic if and only if the function $f={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}}{|{\beta }^{\prime }\left(z\right)|}}$ is nonconstant, and β satisfies the following ODE:

$$\begin{array}{c}\beta {\beta }^{\prime \prime }-2{\beta }^{\prime 2}=0\hfill \end{array}$$

(21)

on the surface.

Here, $c>0$ and $a_i$$(i=1,2,3)$ are constants.

Remark 4.

It is worth noting that there exists at least one nonzero constant $a_i$ ($i=1,\dots ,m$) in Theorem 4 and Corollary 3.

If $a_i=0$ for $i=1,2,\dots ,m$ in Proposition 2, the following theorem provides another main technique for constructing f-biharmonic hypersurfaces:

Theorem 5.

A hypersurface $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}^{m+1},h={\beta }^{-2}({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2))$ with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,a_{m+1})$ in a conformally flat space is proper f-biharmonic if and only if the function $f=c|{\beta }_z{|}^{{\displaystyle \frac{m-4}{2}}}$ is nonconstant, and β satisfies the following PDE:

$$\begin{array}{c}{\displaystyle \sum _{i=1}^m}(\beta {\beta }_{ii}-m{\beta }_i^2)+m(\beta {\beta }_{zz}-2{\beta }_z^2)\hfill \\ +{\displaystyle \frac{(m-2){\displaystyle \sum _{i=1}^m}[{\beta }^2{\beta }_{iiz}-(m-2)\beta {\beta }_i{\beta }_{iz}]}{2{\beta }_z}}+{\displaystyle \frac{(m-2)(m-4){\displaystyle \sum _{i=1}^m}{\beta }^2{\beta }_{iz}^2}{4{\beta }_z^2}}=0\hfill \end{array}$$

(22)

on the hypersurface.

Here, $c>0$ and $a_{m+1}$ are constants.

Proof. 

If $a_i=0$ for $i=1,2,\dots ,m$ in Proposition 2, then ${\xi }_0={\displaystyle \frac{\partial }{\partial z}}$, $H={\xi }_0\left(\beta \right)={\beta }_z$. Substituting these into (14) and simplifying the resulting equation yields (22). Clearly, $f=c|{\beta }_z{|}^{{\displaystyle \frac{m-4}{2}}}$.

By combining these results and applying Proposition 2, the theorem is established. □

As an immediate consequence of Theorem 5, we obtain the following corollary:

Corollary 4.

A surface $\phi :({\mathbb{R}}^2,g={\phi }^{\ast }h)\to ({\mathbb{R}}^3,h={\beta }^{-2}({\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2))$ defined by $\phi (x_1,x_2)=(x_1,x_2,a_3)$ in a conformally flat space is proper f-biharmonic if and only if the function $f={\displaystyle \frac{c}{|{\beta }_z|}}$ is nonconstant, and β satisfies the following PDE:

$$\begin{array}{c}{\displaystyle \sum _{i=1}^2}(\beta {\beta }_{ii}-2{\beta }_i^2)+2(\beta {\beta }_{zz}-2{\beta }_z^2)=0\hfill \end{array}$$

(23)

on the surface.

Here, $c>0$ and $a_3$ are constants.

3. $\mathit{f}$-Biharmonic Surfaces and Biharmonic Conformal Immersions

In this section, we use f-biharmonic surfaces to construct biharmonic conformal immersions, and we conversely construct f-biharmonic surfaces using given conformal immersions of surfaces. We also obtain a classification result for f-biharmonic surfaces in 3-space forms.

From Theorem 2.3 in [3], a map $\varphi :(M^2,g)\to (N^n,h)$ is an f-biharmonic map if and only if $\varphi :(M^2,f^{-1}g)\to (N^n,h)$ is a biharmonic map. The interesting link among biharmonicity, f-biharmonicity and conformality from a 2-manifold can be stated as follows:

Lemma 1

([3]). A map $\varphi :(M^2,g)\to (N^n,h)$ is an f-biharmonic map if and only if $\varphi :(M^2,\overline{g}=f^{-1}g)\to (N^n,h)$ is a biharmonic map. In particular, a surface (i.e., an isometric immersion) $\varphi :(M^2,g={\varphi }^{\ast }h)\to (N^n,h)$ is an f-biharmonic surface if and only if the conformal immersion $\varphi :(M^2,\overline{g}=f^{-1}g)\to (N^n,h)$ is a biharmonic map with the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$.

Proof. 

The first part of the lemma is Theorem 2.3 in [3]. The second part follows from the first part together with the relation for the induced metric ${\varphi }^{\ast }h=g={\lambda }^2\overline{g}=f\overline{g}$, which implies ${\lambda }^2=f$ and hence $\lambda =f^{{\displaystyle \frac{1}{2}}}$. This is Corollary 3.6 in [3]. □

Applying Corollary 3, we have the following proposition which provides proper f-biharmonic surfaces in a conformally flat 3-space.

Proposition 3.

Let $a_1,a_2,a_3$, $c_1,c_2$ and c be positive constants, let

$${\mathbb{R}}_{+}^3=\{(x_1,x_2,z):z>0\}$$

denote the upper half-space, and let $\beta ={\displaystyle \frac{1}{c_{1}z+c_2}}$.

Then, for the function $f={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}{(c_1{a}_{1}x+c_1{a}_{2}y+c_1{a}_3+c_2)}^2}{c_1}}$, the surface

$$\phi :({\mathbb{R}}^2,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{+}^3,h={\beta }^{-2}\left(z\right)\left[{\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2\right]\right)$$

with $\phi (x_1,x_2)=(x_1,x_2,{\displaystyle \sum _{i=1}^2}{a}_i{x}_i+a_3)$ is proper f-biharmonic.

Proof. 

By Corollary 3, if a surface

$$\phi :({\mathbb{R}}^2,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{+}^3,h={\beta }^{-2}\left(z\right)\left[{\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2\right]\right)$$

with $\phi (x_1,x_2)=(x_1,x_2,{\displaystyle \sum _{i=1}^2}{a}_i{x}_i+a_3)$ is proper f-biharmonic, then $\beta \left(z\right)$ satisfies the equation $\beta {\beta }^{\prime \prime }\left(z\right)-2{\beta }^{\prime 2}\left(z\right)=0$ on the surface. The general solution to this equation is $\beta ={\displaystyle \frac{1}{c_{1}z+c_2}}$, where $c_1$ and $c_2$ are constants. To satisfy the requirements of the proposition (i.e., $\beta >0$ on ${\mathbb{R}}_{+}^3$), we take $c_1>0$ and $c_2>0$. From this, the proposition follows. □

Remark 5.

As shown in the proof of Proposition 3, the function β is fully solved from the ODE $\beta {\beta }^{\prime \prime }\left(z\right)-2{\beta }^{\prime 2}\left(z\right)=0$. Hence, for the given isometric immersion (in fact, a graph) in Proposition 3, the conformal factor β related to the conformally flat metric of the ambient space is explicitly given.

By Corollary 4, we can also obtain proper f-biharmonic surfaces.

Proposition 4.

Let $c_1,c_2,c_3,c_4$, $a_3$ and c be positive constants, and let

$${\mathbb{R}}_{+}^3=\{(x_1,x_2,z):z>0\}$$

denote the upper half-space. For each $i=1,2$, with $f={\displaystyle \frac{c{(c_3{a}_3+c_4)}^2(c_1{x}_i+c_2)}{c_3}}$ and $\beta ={\displaystyle \frac{1}{(c_1{x}_i+c_2)(c_{3}z+c_4)}}$ corresponding to i, respectively, then the surface $\phi :({\mathbb{R}}^2,{\phi }^{\ast }h)\to ({\mathbb{R}}_{+}^3,h={\beta }^{-2}({\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2))$ with $\phi (x_1,x_2)=(x_1,x_2,a_3)$ in the conformally flat space is proper f-biharmonic.

Proof. 

We will look for special solutions to Equation (23) of the form $\beta =p\left(x_i\right)q\left(z\right)$ that are nonconstant on the target manifold.

Substituting this expression into Equation (23) and rearranging terms, we obtain the following equation:

$$q^2\left(z\right)\left[p\left(x_i\right)p^{\prime \prime }\left(x_i\right)-2p^{\prime 2}\left(x_i\right)\right]+2p^2\left(x_i\right)\left[q\left(z\right)q^{\prime \prime }\left(z\right)-2q^{\prime 2}\left(z\right)\right]=0.$$

(24)

We find that by introducing a separation constant $c_0$, this equation can be separated into two independent ordinary differential equations. To simplify the analysis and obtain explicit special solutions, we set $c_0=0$, which reduces the system to

$$p\left(x_i\right)p^{\prime \prime }\left(x_i\right)-2p^{\prime }{\left(x_i\right)}^2=0,q\left(z\right)q^{\prime \prime }\left(z\right)-2q^{\prime }{\left(z\right)}^2=0.$$

(25)

Solving (25), we obtain

$$\begin{array}{c}p\left(x_i\right)={(c_1{x}_i+c_2)}^{-1},q\left(z\right)={(c_{3}z+c_4)}^{-1},\hfill \end{array}$$

(26)

where $c_1,c_2,c_3,c_4$ are constants. To satisfy the requirements of the proposition (i.e., $\beta >0$ on ${\mathbb{R}}_{+}^3$), we take $c_1,c_2,c_3,c_4>0$.

Combining these results and applying Corollary 4, we obtain the proposition. □

By Lemma 1 and Proposition 3, we can construct proper biharmonic conformal immersions into a conformally flat space.

Corollary 5.

With the same notations as in Proposition 3, for the positive function $f={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}{(c_1{a}_{1}x+c_1{a}_{2}y+c_1{a}_3+c_2)}^2}{c_1}}$, the conformal immersion

$$\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{\ast }h)\to \left({\mathbb{R}}^3,h={(c_{1}z+c_2)}^2\left[{\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2\right]\right)$$

with $\phi (x_1,x_2)=(x_1,x_2,{\displaystyle \sum _{i=1}^2}{a}_i{x}_i+a_3)$ is proper biharmonic.

Using Lemma 1 and Proposition 4, we can also give proper biharmonic conformal immersions into a conformally flat space.

Corollary 6.

With the same notations as in Proposition 4, for $i=1,2$, $f={\displaystyle \frac{c{(c_3{a}_3+c_4)}^2(c_1{x}_i+c_2)}{c_3}}$ and $\beta ={\displaystyle \frac{1}{(c_1{x}_i+c_2)(c_{3}z+c_4)}}$, respectively, the conformal immersion

$$\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{\ast }h)\to ({\mathbb{R}}^3,h={\beta }^{-2}({\displaystyle \sum _{i=1}^2}d{x}_i^2+d{z}^2))$$

with $\phi (x_1,x_2)=(x_1,x_2,a_3)$ into the conformally flat space is proper biharmonic.

On the other hand, we can use Lemma 1 to construct proper f-biharmonic surfaces. Note that Statement (iii) of Theorem 2.6 in [24] reads as

Proposition 5.

Let $\beta ={\displaystyle \frac{2r^3}{-2rz-2(r^2+z^2)\arctan \frac{z}{r}+k{r}^3(r^2+z^2)}}$, where $r=\sqrt{1+x^2+y^2}$ and $k\ge 6$ is a constant. For $f={\displaystyle \frac{k^2{(1+x^2+y^2)}^2}{4}}$, the conformal immersion $\phi :(S^2\setminus \left\{N\right\},g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1+x^2+y^2)}^2}})\to (S^3\setminus \left\{N^{\prime }\right\},h={\beta }^{-2}{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}})$ with $\phi (x,y)=(x,y,0)$ into the conformal 3-sphere is proper biharmonic.

Using Lemma 1 and Proposition 5, we obtain a family of proper f-biharmonic surfaces into the conformal 3-sphere.

Corollary 7.

Let $\beta ={\displaystyle \frac{2r^3}{-2rz-2(r^2+z^2)\arctan \frac{z}{r}+k{\mathbb{R}}^3(r^2+z^2)}}$, where $r=\sqrt{1+x^2+y^2}$ and $k\ge 6$ is a constant. For $f={\displaystyle \frac{k^2{(1+x^2+y^2)}^2}{4}}$, the surface $\phi :({\mathbb{R}}^2,{\phi }^{\ast }h=k^2(d{x}^2+d{y}^2))\to (S^3\setminus \left\{N^{\prime }\right\},h={\beta }^{-2}{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}})$ with $\phi (x,y)=(x,y,0)$ into the conformal 3-sphere is proper f-biharmonic.

At the end of this section, we present a classification result for f-biharmonic surfaces in 3-dimensional space forms.

Theorem 6.

A surface $\phi :(M^2,g)\to N^3\left(C\right)$ of nonzero constant mean curvature in a 3-space form $N^3\left(C\right)$ with constant sectional curvature C is proper f-biharmonic if and only if $C=0$, and it is a part of a circular cylinder of radius ${\displaystyle \frac{1}{2\left|H\right|}}$ in ${\mathbb{R}}^3$, where H denotes the mean curvature of the surface. Furthermore, no proper CMC f-biharmonic surfaces exist in ${\mathbb{S}}^3$ and ${\mathbb{H}}^3$.

Proof. 

Suppose that a surface with nonzero constant mean curvature H in a 3-space form $N^3\left(C\right)$ (of constant sectional curvature C) is f-biharmonic. By Theorem 2, we obtain

$$\Delta f=\left(\right|A{|}^2-2C)f,A\left(\mathrm{grad}f\right)=0.$$

(27)

To solve Equation (27), we choose an orthonormal frame $\{e_1,e_2,\xi \}$ on $N^3\left(C\right)$ adapted to the surface $M^2$, such that $A_{\xi }\left(e_i\right)={\lambda }_i{e}_i$, where ${\lambda }_i$ represents the principal curvatures in the direction $e_i$, respectively. The second equation of (27) then reduces to

$${\lambda }_1{e}_1\left(f\right)=0,{\lambda }_2{e}_2\left(f\right)=0.$$

(28)

By the assumption $H\ne 0$, we have ${\lambda }_1^2+{\lambda }_2^2\ne 0$. Without loss of generality, assume ${\lambda }_1\ne 0$. Since $f\ne \mathrm{constant}$, Equation (28) yields

$${\lambda }_2=0\mathrm{and}{e}_1\left(f\right)=0.$$

(29)

On the other hand, the Gauss equation gives

$${\mathrm{R}}^M(e_1,e_2,e_1,e_2)={\mathrm{R}}^N(e_1,e_2,e_1,e_2)+{\lambda }_1{\lambda }_2.$$

(30)

Combining this with ${\lambda }_2=0$, we derive the key relation:

$$C=K^{M^2}={\mathrm{R}}^M(e_1,e_2,e_1,e_2)={\mathrm{R}}^N(e_1,e_2,e_1,e_2),$$

(31)

where $K^{M^2}$ denotes the Gauss curvature of $M^2$.

Applying the Codazzi equation, we obtain

$$e_1\left({\lambda }_2\right)=({\lambda }_1-{\lambda }_2)⟨{\nabla }_{e_2}{e}_1,e_2⟩,e_2\left({\lambda }_1\right)=({\lambda }_2-{\lambda }_1)⟨{\nabla }_{e_1}{e}_2,e_1⟩.$$

(32)

Given ${\lambda }_2=0$ and ${\lambda }_1=2H$ (a nonzero constant), this implies

$$⟨{\nabla }_{e_1}{e}_1,e_2⟩=⟨{\nabla }_{e_2}{e}_2,e_1⟩=0.$$

(33)

We now compute ${\mathrm{R}}^M(e_1,e_2,e_1,e_2)$ using the curvature formula for the surface $M^2$:

$${\mathrm{R}}^M(e_1,e_2,e_1,e_2)=e_2⟨{\nabla }_{e_1}{e}_1,e_2⟩+e_1⟨{\nabla }_{e_2}{e}_2,e_1⟩-{⟨{\nabla }_{e_1}{e}_1,e_2⟩}^2-{⟨{\nabla }_{e_2}{e}_2,e_1⟩}^2.$$

(34)

Substituting (33) into (34) and using the relation (31), we find

$$C={\mathrm{R}}^N(e_1,e_2,e_1,e_2)={\mathrm{R}}^M(e_1,e_2,e_1,e_2)=e_2\left(0\right)+e_1\left(0\right)-0^2-0^2=0.$$

This shows that the target space $N^3\left(C\right)$ must be the Euclidean space ${\mathbb{R}}^3$. Combining the above results with Lemma 1 and Theorem 2.10 in [20], we conclude that a surface $(M^2,g)$ in ${\mathbb{R}}^3$ is proper f-biharmonic if and only if it is a part of a circular cylinder of radius ${\displaystyle \frac{1}{2\left|H\right|}}$ in ${\mathbb{R}}^3$.

This completes the proof of the theorem. □

Classical results indicate that circular cylinders in ${\mathbb{R}}^3$ are neither minimal nor biharmonic surfaces. Nevertheless, by Lemma 1 and Theorem 6, such cylinders in Euclidean space can be proper f-biharmonic. We thus obtain the following corollary.

Corollary 8.

Let $\phi :(M^2,g)\to N^3\left(C\right)$ be a surface with nonzero constant mean curvature H in a 3-space form of constant sectional curvature C. Then, the conformal immersion $\phi :(M^2,f^{-1}g)\to N^3\left(C\right)$ with a positive nonconstant conformal factor $f^{1/2}$ is proper biharmonic if and only if $C=0$, and the surface is a part of a circular cylinder of radius ${\displaystyle \frac{1}{2\left|H\right|}}$ in ${\mathbb{R}}^3$.

Remark 6.

In [20], the author proved that no part of the standard sphere $S^2$ can be biharmonically conformally immersed into ${\mathbb{R}}^3$, and a surface of nonzero constant mean curvature can be biharmonically conformally immersed into ${\mathbb{R}}^3$ if and only if it is a part of a circular cylinder—though by a different approach. Our Corollary 8 recovers these results.

Remark 7.

By Corollary 1, a totally umbilical proper f-biharmonic surface $M^2\to N^3$ with the unit normal vector field ξ satisfies ${\mathrm{Ric}}^N(\xi ,\xi )=2H^2>0$, which implies that $N^3$ cannot be a nonpositively curved manifold.

4. Proper $\mathit{f}$-Biharmonic Hypersurfaces ${\mathit{M}}^{\mathit{m}}$ with $\mathit{m}\ge \mathbf{3}$

We have previously noted that the generalized Chen’s conjecture has been disproven by constructing proper biharmonic hyperplanes in a five-dimensional conformally flat space of negative sectional curvature in [12,16]. The authors of these works further utilized this construction and the biharmonicity of product maps to produce numerous examples of proper biharmonic submanifolds in nonpositively curved Riemannian manifolds. Following the idea in [12,16], we investigate the f-biharmonicity of totally umbilical hyperplanes in conformally flat $m+1$-spaces in this section. By seeking special solutions to Equations (19) and (22), we construct examples of totally umbilical proper f-biharmonic hypersurfaces $M^m$ (with $m\ge 3$ and $m\ne 4$) in conformally flat spaces with strictly negative sectional curvature. Furthermore, we provide explicit examples of proper f-biharmonic submanifolds in certain nonpositively curved manifolds.

To simplify and solve Equation (19), we adopt the condition ${\displaystyle \sum _{i=1}^m}{a}_i^2=m$. Under this setting, we can derive explicit solutions, which enables us to construct proper f-biharmonic hypersurfaces in conformally flat spaces with negative sectional curvature. The explicit construction results are presented in the following proposition.

Proposition 6.

Let $c>0$, $a_1$, $\dots ,a_m$ and $a_{m+1}$ be constants with ${\displaystyle \sum _{i=1}^m}{a}_i^2=m$, and let

$${\mathbb{R}}_{+}^{m+1}=\{(x_1,x_2,\dots ,x_m,z):z>0\}$$

denote the upper half-space. Then,

(i) For $\beta =z^{-1}$ and $f=c{(m+1)}^{{\displaystyle \frac{4-m}{4}}}{[{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1}]}^{4-m}$ (with $m\ne 4$), the hypersurface

$$\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{+}^{m+1},h={\beta }^{-2}\left(z\right)\left[{\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2\right]\right)$$

with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is proper f-biharmonic;

(ii) For $\beta =z^{{\displaystyle \frac{m^2-2m}{m^2+4}}}$ and $f=c{[{\displaystyle \frac{{(m^2-2m)}^2}{(m+1){(m^2+4)}^2}}]}^{{\displaystyle \frac{m-4}{4}}}{[{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1}]}^{{\displaystyle \frac{(4-m)(m+2)}{m^2+4}}}$ (with $m\ge 3$ and $m\ne 4$), the hypersurface

$$\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{+}^{m+1},h={\beta }^{-2}\left(z\right)\left[{\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2\right]\right)$$

with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$ is proper f-biharmonic.

Proof. 

We seek special solutions to Equation (19) of the form $\beta =z^t$. We also suppose that ${\displaystyle \sum _{i=1}^m}{a}_i^2=m$, and hence $k_m=1/\sqrt{1+{\displaystyle \sum _{i=1}^m}{a}_i^2}=\sqrt{{\displaystyle \frac{1}{m+1}}}$. Substituting these into (19) and simplifying the resulting expression yields

$$\begin{array}{c}{\displaystyle \frac{[(m^2+4)t^2+(2m+4)t+2m-m^2]m{t}^2{z}^{4t-4}}{4(m+1)}}=0,\hfill \end{array}$$

(35)

which implies

$$\begin{array}{c}(m^2+4)t^2+(2m+4)t+2m-m^2=0.\hfill \end{array}$$

(36)

Solving (36), we find two solutions: $t=-1$ or $t={\displaystyle \frac{m^2-2m}{m^2+4}}$. Thus, the special solutions to (19) are $\beta =z^{-1}$ or $\beta =z^{{\displaystyle \frac{m^2-2m}{m^2+4}}}$ (for $m\ge 3$). Upon further computation, we obtain

$$\begin{array}{c}f=c{k}_m^{{\displaystyle \frac{m-4}{2}}}|{\beta }^{\prime }{\left(z\right)|}^{{\displaystyle \frac{m-4}{2}}}=c{(m+1)}^{{\displaystyle \frac{4-m}{4}}}{\left|z\right|}^{4-m},\hfill \end{array}$$

or

$$\begin{array}{c}f=c{k}_m^{{\displaystyle \frac{m-4}{2}}}|{\beta }^{\prime }{\left(z\right)|}^{{\displaystyle \frac{m-4}{2}}}=c{[{\displaystyle \frac{{(m^2-2m)}^2}{(m+1){(m^2+4)}^2}}]}^{{\displaystyle \frac{m-4}{4}}}{\left|z\right|}^{{\displaystyle \frac{(4-m)(m+2)}{m^2+4}}},\hfill \end{array}$$

where $z={\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1}$.

By combining these results with Theorem 4, the proposition follows. □

The following proposition provides additional proper f-biharmonic hypersurfaces in a conformally flat space of negative sectional curvature:

Proposition 7.

Let c, C and $a_{m+1}$ be positive constants, and let ${\mathbb{R}}_{\ast }^{m+1}=\{(x_1,\dots ,x_m,z)\in {\mathbb{R}}^{m+1}:x_1,\dots ,x_m,z>0\}$. Then,

(i) For $\beta ={({\displaystyle \sum _{i=1}^m}{x}_i+a_{m+1}+C)}^{-1}$ and $f=c{({\displaystyle \sum _{i=1}^m}{x}_i+a_{m+1}+C)}^{4-m}$ (with $m\ne 4$), the hypersurface

$$\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{\ast }^{m+1},h={\beta }^{-2}(x_1,\dots ,x_m,z)\left[{\displaystyle \sum _{j=1}^m}d{x}_j^2+d{z}^2\right]\right)$$

with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,a_{m+1})$ is proper f-biharmonic;

(ii) For $\beta ={({\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{{\displaystyle \frac{m^2-2m}{m^2+4}}}$ and $f=c{({\displaystyle \frac{m^2-2m}{m^2+4}})}^{{\displaystyle \frac{m-4}{2}}}{({\displaystyle \sum _{i=1}^m}{x}_i+a_{m+1}+C)}^{{\displaystyle \frac{(4-m)(m+2)}{m^2+4}}}$ (with $m\ge 3$ and $m\ne 4$), the hypersurface

$$\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to \left({\mathbb{R}}_{\ast }^{m+1},h={\beta }^{-2}(x_1,\dots ,x_m,z)\left[{\displaystyle \sum _{j=1}^m}d{x}_j^2+d{z}^2\right]\right)$$

with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,a_{m+1})$ is proper f-biharmonic.

Proof. 

We seek nonconstant special solutions to Equation (22) of the form $\beta ={({\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^t$, where $t\in \mathbb{R}$ and C is a constant. Substituting this into (22) and simplifying the resulting equation yields

$$\begin{array}{c}m\{(m^2+4)t^2+(2m+4)t+2m-m^2\}{({\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{2t-2}=0.\hfill \end{array}$$

(37)

This implies $(m^2+4)t^2+(2m+4)t+2m-m^2=0$, whose solutions are $t=-1$ or $t={\displaystyle \frac{m^2-2m}{m^2+4}}$. A simple computation then gives

$$f=c|{\beta }_z{|}^{{\displaystyle \frac{m-4}{2}}}{|}_{z=a_{m+1}}=c{|{\displaystyle \sum _{i=1}^m}{x}_i+a_{m+1}+C|}^{4-m},$$

or

$$f=c|{\beta }_z{|}^{{\displaystyle \frac{m-4}{2}}}{|}_{z=a_{m+1}}=c{({\displaystyle \frac{m^2-2m}{m^2+4}})}^{{\displaystyle \frac{m-4}{2}}}{|{\displaystyle \sum _{i=1}^m}{x}_i+a_{m+1}+C|}^{{\displaystyle \frac{(4-m)(m+2)}{m^2+4}}}.$$

By combining these results with Theorem 5, the proposition is proven. □

The following two model spaces are of negative sectional curvatures:

Lemma 2.

Both the conformally flat spaces $({\mathbb{R}}_{+}^{m+1},z^{-2{\displaystyle \frac{m^2-2m}{m^2+4}}}[{\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2])$ and $({\mathbb{R}}_{\ast }^{m+1},{({\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{-2{\displaystyle \frac{m^2-2m}{m^2+4}}}[{\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2])$ (with $m\ge 3$) have negative sectional curvatures.

Proof. 

Let $\beta ={(A{\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^t$, where $t={\displaystyle \frac{m^2-2m}{m^2+4}}$, $A\in \{0,1\}$, and $C\ge 0$ is a constant. For $m\ge 3$, we have $0<t={\displaystyle \frac{m^2-2m}{m^2+4}}<1$. Consider the orthonormal frame $e_i=\beta {\displaystyle \frac{\partial }{\partial x_i}}$ (for $i=1,2,\dots ,m$) and $e_{m+1}=\beta {\displaystyle \frac{\partial }{\partial z}}$ on the space $({\mathbb{R}}^{m+1},h={(A{\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{-2t}[{\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2])$ (with $m\ge 3$). Let $X={\displaystyle \sum _{i=1}^{m+1}}{X}_i{e}_i$ and $Y={\displaystyle \sum _{i=1}^{m+1}}{Y}_i{e}_i$ be an orthonormal basis on any plane section at an arbitrary point.

Similar to the calculation of the sectional curvature in Lemma 3.2 in [16] (or Lemma 3.3 in [12]), the sectional curvature of this conformally flat space is given by

$$\begin{array}{c}K(X,Y)=XX\left(\ln \beta \right)+YY\left(\ln \beta \right)-{\nabla }_{X}X\left(\ln \beta \right)-{\nabla }_{Y}Y\left(\ln \beta \right)\hfill \\ +\left(\right|\nabla \ln \beta {|}^2-{\left(X\ln \beta \right)}^2-{\left(Y\ln \beta \right)}^2)\hfill \\ ={\displaystyle \sum _{i,j=1}^{m+1}}(X_i{X}_j+Y_i{Y}_j)\beta {\beta }_{ij}-{\displaystyle \sum _{i=1}^{m+1}}{\beta }_i^2\hfill \\ =\{A^2[{({\displaystyle \sum _{i=1}^m}{X}_i)}^2+{({\displaystyle \sum _{i=1}^m}{Y}_i)}^2]t(t-1)+2A({\displaystyle \sum _{i=1}^m}{X}_i{X}_{m+1}+Y_i{Y}_{m+1})t(t-1)\hfill \\ -A^{2}m{t}^2+(X_{m+1}^2+Y_{m+1}^2)t(t-1)-t^2\}{(A{\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{2t-2},\hfill \end{array}$$

(38)

where $x_{m+1}=z$, ${\beta }_i={\displaystyle \frac{\partial \beta }{\partial x_i}}$ and ${\beta }_{ij}={\displaystyle \frac{{\partial }^2\beta }{\partial x_j\partial x_i}}$ for $i,j=1,2,\dots ,m+1$.

This, together with $A=0$ or $A=1$, simplifies to

$$\begin{array}{c}K(X,Y)=\{(X_{m+1}^2+Y_{m+1}^2)t(t-1)-t^2\}{(z+C)}^{2t-2},\hfill \end{array}$$

or

$$\begin{array}{c}K(X,Y)=\{[{({\displaystyle \sum _{i=1}^{m+1}}{X}_i)}^2+{({\displaystyle \sum _{i=1}^{m+1}}{Y}_i)}^2]t(t-1)-(m+1)t^2\}{({\displaystyle \sum _{i=1}^m}{x}_i+z+C)}^{2t-2}.\hfill \end{array}$$

Since $0<t<1$ and $[{({\displaystyle \sum _{i=1}^{m+1}}{X}_i)}^2+{({\displaystyle \sum _{i=1}^{m+1}}{Y}_i)}^2]>0$, the curly-braced part in both of the above expressions is negative. Meanwhile, $z+C>0$ and ${\sum }_{i=1}^m{x}_i+z+C>0$, which yields $K(X,Y)<0$. Thus, the lemma follows. □

To conclude this section, we present results on the f-biharmonicity of submanifolds in conformally flat spaces with nonpositive sectional curvature.

Corollary 9.

(i) Any totally umbilical f-biharmonic surface in a three-dimensional manifold $N^3$ with nonpositive sectional curvature is minimal. In fact, such a surface is totally geodesic.

(ii) There exist infinitely many proper f-biharmonic m-dimensional submanifolds (with $m\ge 3$ and $m\ne 4$) in nonpositively curved manifolds.

Proof of (i).

For $m=2$, Theorem 3 implies that if a totally umbilical surface $\phi :M^2\to N^3$ (with unit normal vector field $\xi$) is f-biharmonic, then it is either totally geodesic or it satisfies ${\mathrm{Ric}}^N(\xi ,\xi )=2H^2>0$.

Now, consider a totally umbilical f-biharmonic surface in $N^3$ with strictly negative sectional curvature. Since the Ricci curvature of a manifold with strictly negative sectional curvature cannot be positive (as Ricci curvature is a sum of sectional curvatures), the case ${\mathrm{Ric}}^N(\xi ,\xi )=2H^2>0$ is impossible. Thus, the surface must be totally geodesic. For totally geodesic surfaces, the mean curvature $H=0$, so the surface is minimal. This establishes the first assertion. □

Proof of (ii).

For $m\ge 3$ and $m\ne 4$, Proposition 6 (ii), Proposition 7 (ii), and Lemma 2 together provide numerous examples of totally umbilical proper f-biharmonic hypersurfaces in conformally flat spaces with strictly negative sectional curvature.

To construct infinitely many such submanifolds, we proceed as follows:

1. Let $\phi :({\mathbb{R}}^m,g={\phi }^{\ast }h)\to ({\mathbb{R}}_{+}^{m+1},h={\beta }^{-2}\left(z\right)({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2\left)\right)$ be the proper f-biharmonic hypersurface given in Proposition 6 (ii) with $\phi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1})$.

2. Let $\psi :({\mathbb{R}}_{+}^{m+1},h)\to ({\mathbb{R}}_{+}^{m+1}\times {\mathbb{R}}^k,h\oplus h_1)$ be the totally geodesic embedding defined by $\psi (x_1,\dots ,x_m,z)=(x_1,\dots ,x_m,z,0,\dots ,0)$, where k is a positive integer and $({\mathbb{R}}^k,h_1)$ is a Euclidean space.

By Remark 3 in [23], the composition $\varphi =\psi \circ \phi :({\mathbb{R}}^m,{\phi }^{\ast }(h\oplus h_1))\to ({\mathbb{R}}_{+}^{m+1}\times {\mathbb{R}}^k,h\oplus h_1)$ (with $\varphi (x_1,\dots ,x_m)=(x_1,\dots ,x_m,{\displaystyle \sum _{i=1}^m}{a}_i{x}_i+a_{m+1},0,\dots ,0)$) is a proper f-biharmonic submanifold.

By Lemma 2, the conformally flat space $\left({\mathbb{R}}_{+}^{m+1},z^{-2{\displaystyle \frac{m^2-2m}{m^2+4}}}\left({\displaystyle \sum _{i=1}^m}d{x}_i^2+d{z}^2\right)\right)$ has negative sectional curvature. The Euclidean space $({\mathbb{R}}^k,h_1)$ has zero sectional curvature, so their product $({\mathbb{R}}_{+}^{m+1}\times {\mathbb{R}}^k,h\oplus h_1)$ has nonpositive sectional curvature.

Since k can be any positive integer and the constants $a_1,\dots ,a_{m+1}$ in Proposition 6 (ii) are arbitrary (subject to ${\displaystyle \sum _{i=1}^m}{a}_i^2=m$), we obtain infinitely many such submanifolds. This proves the second assertion. □

Remark 8.

(i) We remark that Corollary 9 disproves the generalized Chen’s conjecture for the f-biharmonic case in higher dimensions, serving as a counterexample to the spirit of the conjecture.

(ii) Combining the proof of Corollary 9 with Remark 3 in [23], and taking the second factor to be a manifold of nonpositive sectional curvature in the product manifold ${\mathbb{R}}_{+}^{m+1}\times {\mathbb{R}}^k$, we also construct proper f-biharmonic m-dimensional submanifolds (with $m\ge 3$ and $m\ne 4$) in nonpositively curved manifolds.

(iii) For proper biharmonic hypersurfaces, their existence in different ambient spaces is as follows: in n-dimensional Euclidean space, for $n\le 4$, there are none, and for $n\ge 5$, their existence remains unproven; in n-dimensional hyperbolic space, for $n=3$, there are none, and for $n\ge 4$, their existence remains unproven; in spherical space, proper biharmonic hypersurfaces can be constructed.

For proper f-biharmonic hypersurfaces, they exist in a wider range of ambient spaces. In general, in Euclidean space, spherical space, hyperbolic space and conformally flat spaces, it is relatively easy to construct proper f-biharmonic hypersurfaces by solving the equation satisfied by the function f, and their existence is relatively easy to prove. However, when certain rigidity conditions are added, the situation changes: for example, in three-dimensional spherical space and hyperbolic space, there are no proper f-biharmonic hypersurfaces with non-zero constant mean curvature, and such hypersurfaces only exist in three-dimensional Euclidean space.

5. Conclusions

This paper focuses on f-biharmonic hypersurfaces in conformally flat spaces. We characterize totally umbilical f-biharmonic hypersurfaces in generic Riemannian manifolds, prove that such proper hypersurfaces (without boundaries) in nonpositively curved manifolds are noncompact, and investigate the f-biharmonicity of totally umbilical hyperplanes in conformally flat spaces. We also classify proper f-biharmonic surfaces with nonzero CMC in three-space forms and explore f-biharmonic hypersurfaces in conformally flat n-spaces with strictly negative sectional curvature.

In particular, the classification result in Theorem 6 reveals a distinct rigidity: even with the flexibility introduced by the weight function f, nonzero CMC proper f-biharmonic surfaces are strongly constrained by the ambient geometry, existing only in the flat Euclidean space ${\mathbb{R}}^3$ rather than in ${\mathbb{S}}^3$ or ${\mathbb{H}}^3$.

Corollary 9 provides infinitely many explicit counterexamples to the generalized Chen’s conjecture for f-biharmonic submanifolds case in higher dimensions, and it offers insights into its validity under the f-biharmonic generalization. Additionally, combining the proof of Corollary 9 with related results, we construct infinitely many proper f-biharmonic m-dimensional submanifolds ($m\ge 3$ and $m\ne 4$) in nonpositively curved manifolds. Our results generalize some works by Ou and Tang [13,16]. We point out that in [13,16], any explicit counterexample to the generalized Chen’s conjecture must be a four-dimensional submanifold.

For future directions, we propose studying f-biharmonic submanifolds in non-flat homogeneous spaces and exploring geometric properties associated with different classes of the weight function f.