Biharmonic Conformal Immersions into a 3-Dimensional Conformally Flat Space

Abstract

This paper investigates biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first establish a characterization of such immersions of totally umbilical surfaces into a generic 3-manifold. It is then proved that any biharmonic conformal immersion of a totally umbilical surface into a nonpositively curved 3-manifold is necessarily a conformal minimal immersion. We further examine the biharmonicity of conformal immersions of totally umbilical planes into a conformally flat 3-space and construct explicit examples of such immersions from a 2-sphere (minus a point) into a conformally flat 3-sphere. Finally, the study is extended to biharmonic conformal immersions of Hopf cylinders associated with a Riemannian submersion.

1. Introduction and Preliminaries

Let $(M^m,g)$ and $(N^n,h)$ be two Riemannian manifolds of dimensions m and n, respectively. Recall that a smooth map $\phi :(M^m,g)\to (N^n,h)$ is called a harmonic map if its tension field vanishes identically; i.e., $\tau \left(\phi \right):={\mathrm{Trace}}_g\overline{\nabla }d\phi \equiv 0$ (see, e.g., [1]), where $\overline{\nabla }$ represents the connection on the pull-back bundle ${\phi }^{*}TN$. From the viewpoint of variational theory, harmonic maps are the critical points of the energy functional $E\left(\phi \right)={\displaystyle \frac{1}{2}}{\int }_{\mathrm{\Omega }}{\left|d\phi \right|}^{2}d{v}_g$, whilst biharmonic maps are defined as the critical points of the bienergy functional

$$E^2\left(\phi ,\mathrm{\Omega }\right)={\displaystyle \frac{1}{2}}{\int }_{\mathrm{\Omega }}{\left|\tau \left(\phi \right)\right|}^{2}d{v}_g,$$

where $\mathrm{\Omega }$ is a compact domain of M and $\tau \left(\phi \right)$ is the tension field of $\phi$. The Euler–Lagrange equation of the functional yields the biharmonic map equation [2]:

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

Here, $R^N$ denotes the curvature operator of $(N^n,h)$ defined by

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

It is clear that any harmonic map is a biharmonic map. Thus, we focus on nonharmonic biharmonic maps, which are termed proper biharmonic maps.

We call a submanifold a biharmonic one if the isometric immersion defining it is a biharmonic map. Then the following theorem holds.

Theorem 1

([3]). Let $\phi :M^m\to N^{m+1}$ be a hypersurface with mean curvature vector field $\eta =H\xi$, where H is the mean curvature of the hypersurface. Then φ is biharmonic if and only if

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

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

It is well known that a surface in a 3-sphere $S^3$ defined by the isometric immersion $S^2({\displaystyle \frac{1}{\sqrt{2}}})\to S^3$ is proper biharmonic [4]. For more examples of biharmonic hypersurfaces, we refer the reader to the book [5] and the references therein.

Biharmonic conformal immersions of hypersurfaces into $(N^{m+1},h)$ generalize the notion of biharmonic isometric immersions of hypersurfaces (i.e., biharmonic hypersurfaces). More precisely, suppose that a hypersurface in $(N^{m+1},h)$ is defined by an isometric immersion $\phi :(M^m,g)\to (N^{m+1},h)$ with the induced metric $g={\phi }^{*}h$. We then refer to the conformal immersion of this associated hypersurface into $(N^{m+1},h)$ as a biharmonic conformal immersion of the hypersurface if there exists a function $\lambda :M^m\to (0,+\infty )$ such that the conformal immersion $\phi :(M^m,\overline{g}={\lambda }^{-2}g)\to (N^{m+1},h)$ with conformal factor $\lambda$ is a biharmonic map. Note that $g={\phi }^{*}h={\lambda }^2\overline{g}$; if $\lambda =1$, the conformal immersion reduces to an isometric immersion.

In particular, the biharmonic equation for the conformal immersion of a surface into a 3-dimensional model space can be stated as follows.

Theorem 2

(see, e.g., [6,7,8]). Let $\phi :(M^2,g)\to (N^3,h)$ be an isometric immersion with mean curvature vector field $\eta =H\xi$. Then, the conformal immersion $\phi :(M^2,g_0=f^{-1}g)\to (N^3,h)$ with the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$ is biharmonic if and only if

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

(1)

where ${\mathrm{Ric}}^N:T_{q}N\to T_{q}N$ is the Ricci operator of the ambient space defined by $\langle {\mathrm{Ric}}^N\left(Z\right),W\rangle ={\mathrm{Ric}}^N(Z,W)$, A denotes the shape operator of the surface with respect to the unit normal vector field ξ, and the operators $\Delta ,\mathrm{grad}$ and $|,|$ are taken with respect to the induced metric $g={\phi }^{*}h$ on the surface.

Note that the induced metric of the surface is $g={\phi }^{*}h={\lambda }^2{g}_0=f{g}_0$, with the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$.

It is obvious from Theorems 1 and 2 that if a surface in $(N^3,h)$ is a minimal surface (i.e., $H=0$), then any conformal immersion of the associated surface into $(N^3,h)$ is harmonic and hence biharmonic; if a surface in $(N^3,h)$ is biharmonic, then a conformal immersion of the associated surface into $(N^3,h)$ with any positive constant conformal factor is always biharmonic. Such maps are trivially biharmonic conformal immersions. Therefore, this paper focuses more on conformal immersions that are proper biharmonic with nonconstant conformal factors.

Motivated by the work on harmonic and biharmonic immersions of surfaces [6,7,9,10,11], this paper focuses on the study of biharmonic conformal immersions of Hopf cylinders and 2-spheres into a conformally flat 3-space, which are canonical and representative surfaces in conformal geometry. To date, numerous interesting examples and significant results have been established regarding harmonic and biharmonic immersions from 2-spheres and cylinders, which are summarized as follows:

• Chern–Goldberg [9]: Any harmonic immersion $\varphi :S^2\to (N^n,h)$ must be minimal or, equivalently, a conformal immersion.
• Sacks–Uhlenbeck [11]: Any harmonic map $\varphi :S^2\to (N^n,h)$ with $n\ge 3$ must be a conformal branched minimal immersion.
• No part of the standard sphere $S^2$ can be biharmonically conformally immersed into ${\mathbb{R}}^3$ (see [7]).
• A non-conformal rotationally symmetric map $\phi :(S^2,d{r}^2+{sin}^{2}rd{\theta }^2)\to (N^2,d{\rho }^2+({\rho }^2+C)d{\varphi }^2)$ with $\phi (r,\theta )=(|cot{\displaystyle \frac{r}{2}}|[1+ln(1+tan{\displaystyle \frac{r}{2}})],\theta )$ is a proper biharmonic map defined locally on a 2-sphere (with two singular points), where $C>0$ is a constant (see [12]). Furthermore, given the function $f={\displaystyle \frac{4{tan}^2\frac{r}{2}(1+{tan}^2\frac{r}{2}){(1+2{tan}^2\frac{r}{2})}^{\frac{3}{2}}}{3{tan}^2\frac{r}{2}(1+{tan}^2\frac{r}{2})(2+{tan}^2\frac{r}{2})+1}}$, the non-conformal rotationally symmetric map $\phi :(S^2,f^{-1}(d{r}^2+{sin}^{2}rd{\theta }^2))\to (S^2,d{\rho }^2+{sin}^2\rho d{\varphi }^2)$ defined by $\phi (r,\theta )=({\displaystyle \frac{1}{2}}arccos({sin}^2{\displaystyle \frac{r}{2}}),2\theta )$ is a proper biharmonic map defined locally on a Riemann 2-sphere (with two singular points) from a Riemann 2-sphere to a 2-sphere (see [13]).
• Proper biharmonic conformal immersions of developable surfaces into ${\mathbb{R}}^3$ exist if and only if the surface is a cylinder (see [7,14]).
• Let $\gamma :I\to S^2$ be a circle with radius ${\displaystyle \frac{1}{\kappa }}$ ($\kappa >1$), and let $\phi :\Sigma ={\cup }_{s\in I}{\pi }^{-1}\left(\gamma \left(s\right)\right)\to S^2\times \mathbb{R}$ be the Hopf cylinder with the induced metric $g={\phi }^{*}h$. For positive constants $d_1,d_2$, and the function $f=d_1{e}^{z\sqrt{{\kappa }^2-1}}+d_2{e}^{-z\sqrt{{\kappa }^2-1}}$, the conformal immersion $\phi :(\Sigma ,\overline{g}=f^{-1}{\phi }^{*}h)\to S^2\times \mathbb{R}$ is a proper biharmonic map (see [7]). Note that the Hopf cylinder has a constant mean curvature $H={\displaystyle \frac{\kappa }{2}}$.
• For constants $d_1$ and $d_2$, and the positive function $f=(d_{1}z+d_2){(1+s^2)}^{3/2}$, the conformal immersion of the cylinder into ${\mathbb{R}}^3$, namely, $\phi :(\Sigma ,f^{-1}(d{s}^2+d{z}^2))\to {\mathbb{R}}^3$ with $\phi (s,z)=(ln(\sqrt{1+s^2}+s)+ln4,\sqrt{1+s^2},z)$, is a proper biharmonic map (see [14]).

For a comprehensive overview of basic examples and properties of biharmonic maps, we refer the reader to [6,8,10,15,16,17,18,19,20,21] and the references therein.

With respect to globally defined biharmonic maps from $S^2$, little is known beyond the biharmonic isometric immersion $S^2\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\to S^3$ (see [4])—or its composition with a totally geodesic map from the 3-sphere $S^3$ into another model space. It is therefore of great interest to investigate the existence of proper biharmonic conformal immersions from $S^2$. Building on the result from [7,14] that proper biharmonic conformal immersions of developable surfaces into ${\mathbb{R}}^3$ exist if and only if the surface is a cylinder, we aim to classify or construct biharmonic conformal immersions of Hopf cylinders (arising from a Riemannian submersion) from certain 3-manifolds.

In the existing literature, various results have been established on biharmonic conformal immersions.

For instance, Ou [6] studied a conformal biharmonic immersion of a surface into Euclidean 3-space, while [7] investigated proper biharmonic immersions of Hopf cylinders with constant mean curvature into $S^2\times \mathbb{R}$.

Compared with the above works, in this paper, we study biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first establish a characterization of biharmonic conformal immersions of a totally umbilical surface into a generic 3-manifold (Theorem 3) and then prove that any biharmonic conformal immersion of a totally umbilical surface into a nonpositively curved 3-manifold is a conformal minimal immersion (Corollary 1). Further, we investigate the biharmonicity of conformal immersions of totally umbilical planes into a conformally flat 3-space (Proposition 1) and construct a family of infinitely many proper biharmonic conformal immersions from a 2-sphere into a conformal 3-sphere with a nonconstant conformal factor and one singular point at the north pole (Theorem 4). We succeed in reproducing the example of the biharmonic isometric immersion $S^2\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\to S^3$ (originally found in [4]) via an alternative method (Proposition 2). Finally, we derive the biharmonic equation for conformal immersions of Hopf cylinders associated with a Riemannian submersion (Proposition 4). We subsequently apply this equation to classify biharmonic conformal immersions of such Hopf cylinders into BCV 3-spaces, which include the conformally flat spaces $S^3$, ${\mathbb{R}}^3$, $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$ (Theorem 5 and Corollary 4).

2. Biharmonic Conformal Immersions of Totally Umbilical Surfaces into a 3-Dimensional Conformally Flat Space

In this section, we establish a characterization of biharmonic conformal immersions of totally umbilical surfaces into a generic 3-manifold. As an application, we propose a method to construct proper biharmonic conformal immersions: starting with a totally umbilical surface in a conformally flat space, we then perform an appropriate conformal transformation of the conformally flat metric to another conformally flat metric. We subsequently apply this method to investigate the biharmonicity of a family of conformal immersions and construct proper biharmonic conformal immersions from $S^2$ into a conformally flat 3-sphere. We also provide numerous examples of biharmonic conformal immersions from ${\mathbb{R}}^2$ or $H^2(-1)$.

2.1. Biharmonic Conformal Immersions of Totally Umbilical Surfaces into a Riemannian 3-Manifold

Theorem 3.

Let $\phi :(M^2,g)\to (N^3,h)$ be a totally umbilical surface with mean curvature vector field $\eta =H\xi$. Then, the conformal immersion $\phi :(M^2,f^{-1}g)\to (N^3,h)$ is biharmonic if and only if one of the following cases holds:

(i) 

The surface is totally geodesic;

(ii) 

The surface has nonzero mean curvature H, and f and H satisfy

$$\begin{array}{c}{\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }={\mathrm{grad}}_{g}H,{\mathrm{Ric}}^N(\xi ,\xi )={\left|A\right|}^2=2H^2,{\mathrm{grad}}_g\left(fH\right)=0.\hfill \end{array}$$

(2)

In this case, there exists a nonzero function $f={c\left|H\right|}^{-1}$ on the surface, where $c>0$ is a constant. Moreover, by virtue of the totally umbilical property of the surface, the first equation of (2) holds naturally.

Proof. 

Let $\phi :(M^2,g)\to (N^3,h)$ be a totally umbilical surface with mean curvature vector field $\eta =H\xi$. Let $\{e_1,e_2,\xi \}$ be an orthonormal frame on $(N^3,h)$ adapted to the surface. Then, we have $A\left(e_i\right)=H{e}_i$ for $i=1,2$, ${\left|A\right|}^2=2H^2$ and $A\left({\mathrm{grad}}_{g}H\right)=H{\mathrm{grad}}_{g}H.$

A straightforward computation gives

$$\begin{array}{c}\hfill \sum _{i=1}^2\langle R^N(e_i,e_j)e_i,\xi \rangle =\sum _{i=1}^2{R}^N(\xi ,e_i,e_i,e_j)=-{\mathrm{Ric}}^N(e_j,\xi ).\end{array}$$

(3)

Note that $\left\{e_i\right\}$ ( $i=1,2$) are principal directions with principal curvature H. We can compute that

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

(4)

and

$$\begin{array}{c}\left({\nabla }_{e_j}B\right)(e_i,e_i)=e_j\left(B(e_i,e_i)\right)-B({\nabla }_{e_j}{e}_i,e_i)-B({\nabla }_{e_j}{e}_i,e_i)\hfill \\ =e_j\left(H\right)-2B(e_i,e_i)\langle {\nabla }_{e_j}{e}_i,e_i\rangle =e_j\left(H\right),forj=1,2.\hfill \end{array}$$

(5)

Here, the covariant derivative of the second fundamental form 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 (4) from (5) and utilizing (3) together with the Codazzi equation for a surface, we obtain

$$\begin{array}{c}\hfill e_j\left(H\right)=R^N(\xi ,e_i,e_j,e_i).\end{array}$$

(6)

Summing both sides of (6) over i from 1 to 2 yields

$$e_j\left(H\right)={\mathrm{Ric}}^N(e_j,\xi ),j=1,2,i.e.,{\mathrm{grad}}_{g}H={\left({\mathrm{Ric}}^N\left(\xi \right)\right)}^{\top }.$$

(7)

It should be noted that (7) holds naturally due to the totally umbilical property of the surface.

Note that if $H=0$, then the surface is totally geodesic and hence biharmonic. From this point onward, we consider the case where $H\ne 0$. Substituting (7) into the second equation of (1) and leveraging the totally umbilical property of the surface, we derive

$${\mathrm{grad}}_g\left(fH\right)=0,$$

(8)

which implies that the function $f\left|H\right|=c$ is a positive constant on the surface. We substitute this into the first equation of (1) to obtain

$${\mathrm{Ric}}^N(\xi ,\xi )={\left|A\right|}^2=2H^2.$$

(9)

Clearly, if H is a nonzero constant, then f must also be a constant, and the surface is indeed biharmonic.

Summarizing the above results, we establish the theorem. □

Applying Theorem 3, we derive the following corollary.

Corollary 1.

Any biharmonic conformal immersion of a totally umbilical surface into a nonpositively curved 3-manifold $(N^3,h)$ is a conformal minimal immersion.

Proof. 

From Theorem 3, we directly conclude that if a conformal immersion of a totally umbilical surface (with mean curvature H) into a nonpositively curved manifold $(N^3,h)$ is biharmonic, then the surface must be totally geodesic and thus minimal. This implies that the biharmonic conformal immersion is a conformal minimal immersion. Thus, the corollary is established. □

Remark 1.

By applying Theorem 3 and drawing on existing results (see, e.g., [4,5,16]), we can conclude that if a conformal immersion of a totally umbilical surface into ${\mathbb{R}}^3$,$H^3$ or $S^3$ is biharmonic, then the associated surface must be either a totally geodesic surface or $S^2\left({\displaystyle \frac{1}{\sqrt{2}}}\right)$ in $S^3$, as previously established in [8] by an alternative method.

2.2. Biharmonic Conformal Immersions of Totally Umbilical Planes into a Conformally Flat 3-Space

Now, we are ready to present the method for constructing proper biharmonic conformal immersions.

Proposition 1.

Consider an isometric immersion $({\mathbb{R}}^2,g_0={\phi }^{*}{h}_0)\to ({\mathbb{R}}^3,h_0=F^{-2}(d{x}^2+d{y}^2+d{z}^2))$ with $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$, where $a_1,a_2$ and $a_3$ are constants. Then, the conformal immersion $\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}{h}_0)$ is proper biharmonic if and only if $F\beta$ satisfies the following:

$$\begin{array}{c}{\displaystyle \frac{1+2a_1^2+a_2^2}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{xx}-2{\left({\left(F\beta \right)}_x\right)}^2]+{\displaystyle \frac{1+a_1^2+2a_2^2}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{yy}-2{\left({\left(F\beta \right)}_y\right)}^2]\hfill \\ +{\displaystyle \frac{2+a_1^2+a_2^2}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{zz}-2{\left({\left(F\beta \right)}_z\right)}^2]+{\displaystyle \frac{2a_1{a}_2}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{xy}-2{\left(F\beta \right)}_x{\left(F\beta \right)}_y]\hfill \\ -{\displaystyle \frac{2a_1}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{xz}-2{\left(F\beta \right)}_x{\left(F\beta \right)}_z]-{\displaystyle \frac{2a_2}{1+a_1^2+a_2^2}}[F\beta {\left(F\beta \right)}_{yz}-2{\left(F\beta \right)}_y{\left(F\beta \right)}_z]=0\hfill \end{array}$$

(10)

when restricted to the surface $z=a_{1}x+a_{2}y+a_3$, and

$$f={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}}{|-a_1{\left(F\beta \right)}_x-a_2{\left(F\beta \right)}_y+{\left(F\beta \right)}_z|}}\circ \phi ,$$

(11)

where the constant $c>0$. Furthermore, the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$ and the surface $({\mathbb{R}}^2,{\phi }^{*}h)$ has the mean curvature $H={\displaystyle \frac{-a_1{\left(F\beta \right)}_x-a_2{\left(F\beta \right)}_y+{\left(F\beta \right)}_z}{\sqrt{1+a_1^2+a_2^2}}}$.

Proof. 

First, we present the following claim.

Claim: The surface $\phi :({\mathbb{R}}^2,g={\phi }^{*}{h}_0)\to ({\mathbb{R}}^3,h_0=F^{-2}(d{x}^2+d{y}^2+d{z}^2))$ with $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$ is totally umbilical with the mean curvature $H_0={\displaystyle \frac{-a_1{F}_x-a_2{F}_y+F_z}{\sqrt{1+a_1^2+a_2^2}}}$, where $F:{\mathbb{R}}^3\to (0,+\infty )$ and $a_i$ ($i=1,2,3$) are constants.

Proof of Claim: We can readily verify that the surface $\phi :({\mathbb{R}}^2,{\phi }^{*}{h}^0)\to ({\mathbb{R}}^3,h^0=d{x}^2+d{y}^2+d{z}^2)$ with $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$ is totally geodesic, with the unit normal vector field ${\xi }^0={\displaystyle \frac{-a_1\frac{\partial }{\partial x}-a_2\frac{\partial }{\partial y}+\frac{\partial }{\partial z}}{\sqrt{1+a_1^2+a_2^2}}}$ and the mean curvature $H^0=0$. It follows from the well-known result (see, e.g., [10,17,22]) that $\phi :({\mathbb{R}}^2,g_0={\phi }^{*}{h}_0)\to ({\mathbb{R}}^3,h_0=F^{-2}(d{x}^2+d{y}^2+d{z}^2))$ (where $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$) is totally umbilical with the unit normal vector field ${\xi }_0=F{\xi }^0$ and the mean curvature $H_0=F{H}^0+{\xi }^{0}F={\xi }^{0}F$. Thus, the claim is established.

Subsequently, by the above claim, the surface $\phi :({\mathbb{R}}^2,g={\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}{h}_0)$ (with $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$) is totally umbilical, with the unit normal vector field $\xi =\beta {\xi }_0$ and the mean curvature $H={\displaystyle \frac{-a_1{\left(F\beta \right)}_x-a_2{\left(F\beta \right)}_y+{\left(F\beta \right)}_z}{\sqrt{1+a_1^2+a_2^2}}}$.

A straightforward computation yields (see, e.g., [17])

$$\begin{array}{c}\mathrm{Ric}(\xi ,\xi )={\Delta }_{h}ln\left(F\beta \right)+[{\mathrm{Hess}}_h(ln\left(F\beta \right))(\xi ,\xi )-{(\xi ln\left(F\beta \right))}^2+|{\mathrm{grad}}_{h}ln\left(F\beta \right){|}_h^2]\hfill \\ =\left(F\beta \right){\Delta }_{h^0}\left(F\beta \right)-2{\left|{\mathrm{grad}}_{h^0}\left(F\beta \right)\right|}_{h^0}^2+F\beta {\mathrm{Hess}}_{h^0}\left(F\beta \right)({\xi }^0,{\xi }^0)\hfill \\ ={\displaystyle \frac{1+2a_1^2+a_2^2}{1+a_1^2+a_2^2}}F\beta {\left(F\beta \right)}_{xx}-2{\left({\left(F\beta \right)}_x\right)}^2+{\displaystyle \frac{1+a_1^2+2a_2^2}{1+a_1^2+a_2^2}}F\beta {\left(F\beta \right)}_{yy}-2{\left({\left(F\beta \right)}_y\right)}^2\hfill \\ +{\displaystyle \frac{2+a_1^2+a_2^2}{1+a_1^2+a_2^2}}F\beta {\left(F\beta \right)}_{zz}-2{\left({\left(F\beta \right)}_z\right)}^2+{\displaystyle \frac{2a_1{a}_{2}F\beta {\left(F\beta \right)}_{xy}}{1+a_1^2+a_2^2}}-{\displaystyle \frac{2a_{1}F\beta {\left(F\beta \right)}_{xz}}{1+a_1^2+a_2^2}}-{\displaystyle \frac{2a_{2}F\beta {\left(F\beta \right)}_{yz}}{1+a_1^2+a_2^2}}.\hfill \end{array}$$

By Theorem 3, the conformal immersion $\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}{h}_0)$ is proper biharmonic if and only if $f={c\left|H\right|}^{-1}={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}}{|-a_1{\left(F\beta \right)}_x-a_2{\left(F\beta \right)}_y+{\left(F\beta \right)}_z|}}$ (where $c>0$ is a constant) and $\mathrm{Ric}(\xi ,\xi )=2H^2$ holds on the surface. This implies that Equation (10) holds on the surface.

Thus, the proposition is established. □

Applying Proposition 1, we immediately obtain the following corollaries, which can be used to construct infinitely many examples of proper biharmonic conformal immersions.

Corollary 2.

Consider an isometric immersion $({\mathbb{R}}^2,g_0={\phi }^{*}{h}_0)\to ({\mathbb{R}}^3,h_0=F^{-2}(d{x}^2+d{y}^2+d{z}^2))$ with $\phi (x,y)=(x,y,a_3)$, where $a_3$ is a constant. Then, the conformal immersion $\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}{h}_0)$ is proper biharmonic if and only if $F\beta$ satisfies the following:

$$F\beta {\left(F\beta \right)}_{xx}-2{\left({\left(F\beta \right)}_x\right)}^2+F\beta {\left(F\beta \right)}_{yy}-2{\left({\left(F\beta \right)}_y\right)}^2+2F\beta {\left(F\beta \right)}_{zz}-4{\left({\left(F\beta \right)}_z\right)}^2=0$$

(12)

when restricted to the surface $z=a_3$, and

$$f={\displaystyle \frac{c}{{|\left(F\beta \right)}_z|}}\circ \phi ,$$

(13)

where $c>0$ is a constant. Moreover, the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$ and the surface $({\mathbb{R}}^2,{\phi }^{*}h)$ has the mean curvature $H={\left(F\beta \right)}_z$.

Proof. 

By taking $a_1=a_2=0$ in Proposition 1, we immediately obtain the corollary. □

Corollary 3.

For constants $a_1,a_2$ and $a_3$, the conformal immersion $\phi :({\mathbb{R}}^2,f^{-1}{\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}\left(z\right)(d{x}^2+d{y}^2+d{z}^2))$ with $\phi (x,y)=(x,y,a_{1}x+a_{2}y+a_3)$ is proper biharmonic if and only if $\beta \left(z\right)$ is nonconstant and satisfies the following equation:

$$\beta {\beta }^{\prime \prime }-2{\beta }^{\prime 2}=0$$

(14)

when restricted to the surface $z=a_{1}x+a_{2}y+a_3$, and

$$f={\displaystyle \frac{c\sqrt{1+a_1^2+a_2^2}}{|{\beta }^{\prime }\left(z\right)|}}\circ \phi ,$$

where $c>0$ is a constant. Moreover, the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$ and the surface $({\mathbb{R}}^2,{\phi }^{*}h)$ has the mean curvature $H={\displaystyle \frac{{\beta }^{\prime }\left(z\right)}{\sqrt{1+a_1^2+a_2^2}}}$.

Proof. 

By applying Proposition 1 with $F=1$ and $\beta =\beta \left(z\right)$ (depending only on z), we immediately derive the corollary. □

By taking $a_1=a_2=1$, $a_3=0$, $c={\displaystyle \frac{\sqrt{3}}{3}}$ and $\beta \left(z\right)={\displaystyle \frac{1}{z}}$ in Corollary 3, we obtain the following example immediately.

Example 1.

The conformal immersion $\phi :({\mathbb{R}}_{+}^2,2(d{x}^2+dxdy+d{y}^2))\to ({\mathbb{R}}_{+}^3,z^2(d{x}^2+d{y}^2+d{z}^2))$ defined by $\phi (x,y)=(x,y,x+y)$ is proper biharmonic with conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}=x+y$. Here, ${\mathbb{R}}_{+}^2=\{(x,y)\in {\mathbb{R}}^2,x,y>0\}$ and ${\mathbb{R}}_{+}^3=\{(x,y,z)\in {\mathbb{R}}^3,x,y,z>0\}$. Note that the surface $({\mathbb{R}}_{+}^2,{\phi }^{*}h)$ has nonconstant mean curvature $H=-{\displaystyle \frac{1}{\sqrt{3}{(x+y)}^2}}$ and the domain surface $({\mathbb{R}}_{+}^2,2(d{x}^2+dxdy+d{y}^2))$ is flat.

2.3. Biharmonic Conformal Immersions of a 2-Sphere into a Conformal 3-Sphere

Let us now consider a totally geodesic immersion defined by

$$\phi :(S^2,g_0)\to (S^3,h_0),\phi \left(u\right)=(u,0),\forall u\in S^2\subset {\mathbb{R}}^3$$

(15)

from the standard 2-sphere into the standard 3-sphere.

It would be interesting to know if there is a nonconstant positive function $\beta$ on the target sphere $S^3$ such that the conformal immersion $\phi :(S^2,g_0)\to (S^3,{\beta }^{-2}{h}_0)$ from the standard 2-sphere into the conformal 3-sphere is a proper biharmonic map. Note that the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}={\beta }^{-1}\circ \phi$.

For the calculations, it is convenient to use local coordinates on the domain and target spheres. In local coordinates, the map (15) can be described as

$$\begin{array}{c}\phi :(S^2\setminus \left\{N\right\},g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1+x^2+y^2)}^2}})\to (S^3\setminus \left\{N^{\prime }\right\},h_0={\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}}),\hfill \\ \phi (x,y)=(x,y,0),\hfill \end{array}$$

where $\left\{N\right\}$ and $\left\{N^{\prime }\right\}$ denote the north poles on the domain and target spheres, respectively.

We aim to find a positive function $\beta$ on the target sphere $S^3$ such that the conformal immersion given by

$$\begin{array}{c}\phi :(S^2\setminus \left\{N\right\},g_0)\to (S^3\setminus \left\{N^{\prime }\right\},h={\beta }^{-2}{h}_0),\phi (x,y)=(x,y,0)\hfill \end{array}$$

is proper biharmonic.

In what follows, we define the function ${\mathrm{\Phi }}_k(x,y,z)$ by

$$\begin{array}{c}{\mathrm{\Phi }}_k(x,y,z)={\displaystyle \frac{2r^3}{-2rz-2(r^2+z^2)arctan\frac{z}{r}+k{r}^3(r^2+z^2)}}\hfill \end{array}$$

(16)

where k is a constant and $r=\sqrt{1+x^2+y^2}$.

Note that for $k\ge 6$, the function $-2rz-2(r^2+z^2)arctan{\displaystyle \frac{z}{r}}+k{r}^3(r^2+z^2)\ge -2rz+(r^2+z^2)(6r^3-\pi )>{(r-z)}^2+(r^2+z^2)\ge 1$, and hence ${\mathrm{\Phi }}_k(x,y,z)>0$ when $k\ge 6$.

In addition, we define the function ${\psi }_w(x,y,z)$ by

$$\begin{array}{c}{\psi }_w(x,y,z)={\displaystyle \frac{r^3(r^2+z^2)}{-2rz-2(r^2+z^2)arctan\frac{z}{r}+w(x,y)r^3(r^2+z^2)}},\hfill \end{array}$$

(17)

where $r=\sqrt{1+x^2+y^2}$ and $w=w(x,y)$ is a nonzero differentiable function.

Our next theorem provides a classification of biharmonic maps in a family of conformal immersions of the standard 2-sphere into the conformal 3-sphere, as well as a family of proper biharmonic conformal immersions defined on $S^2\setminus \left\{N\right\}$.

Theorem 4.

(i) Any proper biharmonic conformal immersion $\phi :(S^2,g_0)\to (S^3,h={\beta }^{-2}{h}_0)$ from the standard 2-sphere into the conformal 3-sphere with $\phi \left(u\right)=(u,0),\forall u\in S^2\subset {\mathbb{R}}^3$, is actually defined on the 2-sphere minus at least one point.

(ii) Let ${\psi }_w(x,y,z)$ be defined by (17), and let $w=w(x,y)$ be a positive harmonic function such that ${\psi }_w(x,y,z)>0$ on ${\mathbb{R}}^3$. For $\beta ={\displaystyle \frac{2{\psi }_w(x,y,z)}{1+x^2+y^2+z^2}}$, the conformal immersion $\phi :(S^2\setminus \left\{N\right\},g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1+x^2+y^2)}^2}})\to (S^3\setminus \left\{N^{\prime }\right\},h={\beta }^{-2}{h}_0={\beta }^{-2}{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}})$ with $\phi (x,y)=(x,y,0)$ into the conformal 3-sphere is proper biharmonic with the conformal factor $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{w(x,y)(1+x^2+y^2)}{2}}$.

(iii) For the family of positive functions $\beta ={\mathrm{\Phi }}_k(x,y,z)$ defined by (16) with $k\ge 6$, the conformal immersion $\phi :(S^2\setminus \left\{N\right\},g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1+x^2+y^2)}^2}})\to (S^3\setminus \left\{N^{\prime }\right\},h={\beta }^{-2}{h}_0={\beta }^{-2}{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}})$ with $\phi (x,y)=(x,y,0)$ into the conformal 3-sphere is proper biharmonic with the conformal factor $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{k(1+x^2+y^2)}{2}}$.

Proof. 

First, one can readily verify that the immersion $\phi :({\mathbb{R}}^2,g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1+x^2+y^2)}^2}})\to ({\mathbb{R}}^3,h_0=F^{-2}(d{x}^2+d{y}^2+d{z}^2)={\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}})$ defined by $\phi (x,y)=(x,y,0)$ is a totally umbilical surface, with the unit normal vector field ${\xi }_0=F{\displaystyle \frac{\partial }{\partial z}}$ and the mean curvature $H_0{|}_{z=0}=F_z{{|}_{z=0}=z|}_{z=0}=0$, where $F={\displaystyle \frac{1+x^2+y^2+z^2}{2}}$.

By Corollary 2, we can conclude that the totally umbilical surface $\phi :({\mathbb{R}}^2,g={\phi }^{*}h)\to ({\mathbb{R}}^3,h={\beta }^{-2}{h}_0)$ with $\phi (x,y)=(x,y,0)$ has the unit normal vector field $\xi =F\beta {\displaystyle \frac{\partial }{\partial z}}$ and the mean curvature $H={\left(F\beta \right)}_z$.

Next, we aim to find the functions f and $\beta$ such that $f^{-1}{\phi }^{*}h=g_0$ and hence ${\displaystyle \frac{{\left(F\beta \right)}_z}{c{\left(F\beta \right)}^2}}=\pm F^{-2}$. Taking $c=1$, we now consider the following PDE:

$$\begin{array}{c}{\displaystyle \frac{{\left(F\beta \right)}_z}{{\left(F\beta \right)}^2}}=F^{-2},\hfill \end{array}$$

(18)

where $F={\displaystyle \frac{1+x^2+y^2+z^2}{2}}$.

Equation (18) has the general solution given by

$$\begin{array}{c}F\beta ={\psi }_w(x,y,z)={\displaystyle \frac{r^3(r^2+z^2)}{-2rz-2(r^2+z^2)arctan\frac{z}{r}+w(x,y)r^3(r^2+z^2)}},\hfill \end{array}$$

(19)

where $r=\sqrt{1+x^2+y^2}$ and $w=w(x,y)$ is a nonzero differentiable function.

We proceed to show that $w(x,y)$ is a nonzero harmonic function on ${\mathbb{R}}^2$; i.e., $w_{xx}+w_{yy}=0$. In fact, substituting $F\beta ={\psi }_w(x,y,z)$ into the left-hand side of (12) and restricting it to the surface $z=0$, we conclude that

$$\begin{array}{c}\{F\beta {\left(F\beta \right)}_{xx}-2{\left({\left(F\beta \right)}_x\right)}^2+F\beta {\left(F\beta \right)}_{yy}-2{\left({\left(F\beta \right)}_y\right)}^2+2F\beta {\left(F\beta \right)}_{zz}-4{\left({\left(F\beta \right)}_z\right)}^2\}{|}_{z=0}\hfill \\ =\{{\psi }_w{\left({\psi }_w\right)}_{xx}-2{\left({\psi }_w\right)}_x^2+{\psi }_w{\left({\psi }_w\right)}_{yy}-2{\left({\psi }_w\right)}_y^2+2{\psi }_w{\left({\psi }_w\right)}_{zz}-4{\left({\psi }_w\right)}_z^2\}{|}_{z=0}=-{\displaystyle \frac{w_{xx}+w_{yy}}{w^3}}.\hfill \end{array}$$

(20)

Using this and (12), it follows that $w_{xx}+w_{yy}=0$, and hence $w(x,y)$ is a nonzero harmonic function on ${\mathbb{R}}^2$. This implies that $\beta ={\displaystyle \frac{2{\psi }_w(x,y,z)}{1+x^2+y^2+z^2}}$ and $f^{{\displaystyle \frac{1}{2}}}={\left({\left(F\beta \right)}_z{|}_{z=0}\right)}^{-{\displaystyle \frac{1}{2}}}={\left({\left(F\beta \right)}^2{F}^{-2}{|}_{z=0}\right)}^{-{\displaystyle \frac{1}{2}}}={\beta }^{-1}\circ \phi ={\displaystyle \frac{w(x,y)(1+x^2+y^2)}{2}}$, and hence $w(x,y)>0$.

We next show that $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{w(x,y)(1+x^2+y^2)}{2}}$ is locally defined on $S^2$. Suppose, for a contradiction, that $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{w(x,y)(1+x^2+y^2)}{2}}$ is globally defined on $S^2$ (where $S^2\equiv {\mathbb{R}}^2\cup \{\infty \}$). This implies that the positive function $w(x,y)$ is a nonconstant, bounded function on $S^2$ (i.e., ${\mathbb{R}}^2\cup \{\infty \})$. By Liouville’s theorem, a bounded harmonic function on Euclidean space is constant, so $w(x,y)$ must be constant, which is a contradiction. Therefore, the conformal factor $\lambda =f^{{\displaystyle \frac{1}{2}}}$ has at least one singular point at the north pole.

Finally, we show that there must exist a positive harmonic function $w(x,y)$ on ${\mathbb{R}}^2$ such that each of the two functions $f^{{\displaystyle \frac{1}{2}}}$ and $\beta$ has exactly one singular point at the north pole. As we have mentioned, for any constant $k\ge 6$, the function ${\mathrm{\Phi }}_k(x,y,z)$ defined by (16) is positive. For the constant harmonic function $w=w(x,y)=k\ge 6$, we immediately obtain the two positive functions $\beta ={\displaystyle \frac{{\psi }_k(x,y,z)}{F}}={\mathrm{\Phi }}_k(x,y,z)$ on ${\mathbb{R}}^3$ and $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{k(1+x^2+y^2)}{2}}$ on ${\mathbb{R}}^2$, each of which has exactly one singular point at the north pole.

Summarizing the above results, we complete the proof. □

Remark 2.(1) We note that none of the locally defined proper biharmonic conformal immersions from the standard $S^2$ given in Theorem 4 can be extended to a global map $\phi :(S^2,g_0)\to (S^3,h={\beta }^{-2}{h}_0)$. This is because each such a conformal immersion has at least one singular point at the north pole. In particular, each of the biharmonic conformal immersions in part (iii) of Theorem 4 has exactly one singular point at the north pole.

It is interesting to compare the examples of proper biharmonic conformal immersions provided in part (iii) of Theorem 4 with the well-known theorem by Sacks–Uhlenbeck [11], which states that any harmonic map $\varphi :S^2\to (N^n,h)$ (with $n\ge 3$) must be a conformal minimal immersion away from points where the differential of the map vanishes.

(2) By Corollary 1, the conformal 3-sphere $(S^3,h={\beta }^{-2}{h}_0)$ in Theorem 4 cannot be a nonpositively curved manifold.

We now present a description of the biharmonic isometric immersion $S^2\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\to S^3$ in local coordinates.

Proposition 2.

A (part of) sphere $S^2({\displaystyle \frac{1}{\sqrt{2}}})$ in $S^3$ is proper biharmonic. Furthermore, the isometric immersion can be locally expressed as

$$\begin{array}{c}\phi :\left(S^2({\displaystyle \frac{1}{\sqrt{2}}}),{\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(2+x^2+y^2)}^2}}\right)\to \left(S^3,{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}}\right),\phi (x,y)=(x,y,1).\hfill \end{array}$$

(21)

Proof. 

By applying Corollary 2 with $F={\displaystyle \frac{1+x^2+y^2+z^2}{2}}$ and $\beta =1$, and using an argument analogous to that in Theorem 4, we conclude that the surface defined by the isometric immersion

$$\begin{array}{c}\phi :\left(S^2({\displaystyle \frac{1}{\sqrt{2}}})\setminus \left\{N\right\},{\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(2+x^2+y^2)}^2}}\right)\to \left(S^3\setminus \left\{N^{\prime }\right\},{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1+x^2+y^2+z^2)}^2}}\right),\phi (x,y)=(x,y,1)\hfill \end{array}$$

(22)

is proper biharmonic. Since the conformal factor $f^{{\displaystyle \frac{1}{2}}}={\left({\left(F\beta \right)}_z{|}_{z=1}\right)}^{-{\displaystyle \frac{1}{2}}}=1$, the map $\phi$ as an isometric immersion can be extended smoothly to $S^2({\displaystyle \frac{1}{\sqrt{2}}})$. From this, we establish the proposition. □

Remark 3.

We note that the authors in [4] showed that a (part of) sphere $S^2({\displaystyle \frac{1}{\sqrt{2}}})$ in $S^3$ is proper biharmonic using a different method. Interestingly, our Proposition 2 provides a description of the above proper biharmonic map in local coordinates.

To conclude this section, using an argument analogous to that in Theorem 4, we construct a family of biharmonic conformal immersions from $H^2(-1)$ as follows.

Proposition 3.

Let $\rho =\sqrt{1-x^2-y^2}$ and $\beta ={\displaystyle \frac{2{\rho }^3({\rho }^2-z^2)}{-2\rho z-2({\rho }^2-z^2)arctan\frac{z}{\rho }+w(x,y){\rho }^3({\rho }^2-z^2)}}$, where $w=w(x,y)$ is a positive harmonic function on ${\mathbb{R}}^2$ such that $\beta >0$. Locally, the conformal immersion $\phi :(H^2(-1),g_0={\displaystyle \frac{4(d{x}^2+d{y}^2)}{{(1-x^2-y^2)}^2}})\to (H^3(-1),h={\beta }^{-2}{\displaystyle \frac{4(d{x}^2+d{y}^2+d{z}^2)}{{(1-x^2-y^2-z^2)}^2}})$ with $\phi (x,y)=(x,y,0)$ (from a hyperbolic 2-space into the conformal hyperbolic 3-space) is proper biharmonic, with the conformal factor $f^{{\displaystyle \frac{1}{2}}}={\displaystyle \frac{w(x,y)(1-x^2-y^2)}{2}}$.

3. Biharmonic Conformal Immersions of Hopf Cylinders of a Riemannian Submersion

Let $\pi :(N^3,h)\to (B^2,h_1)$ be a Riemannian submersion with totally geodesic fibres. Suppose $\gamma :I\to (B^2,h_1)$ is an immersed regular curve parametrized by arclength, and let $\{\overline{X}={\gamma }^{\prime },\overline{\xi }\}$ be a Frenet frame along $\gamma$. The Frenet formula for this frame is given by ${\overline{\nabla }}_{\overline{X}}\overline{X}=\overline{\kappa }\overline{\xi },{\overline{\nabla }}_{\overline{X}}\overline{\xi }=-\overline{\kappa }\overline{X}$, where $\overline{\nabla }$ denotes the Levi-Civita connection of $(B^2,h_1)$ and $\overline{\kappa }$ stands for the geodesic curvature of $\gamma$. Accordingly, $\Sigma ={\cup }_{s\in I}{\pi }^{-1}\left(\gamma \left(s\right)\right)$ defines a surface in $N^3$, which is called a Hopf cylinder and may alternatively be regarded as a disjoint union of all horizontal lifts of the curve $\gamma$.

Let $\tilde{\gamma }:I\to (N^3,h)$ be a horizontal lift of $\gamma$, with X and $\xi$ denoting the horizontal lifts of $\overline{X}$ and $\overline{\xi }$, respectively. We can construct an orthonormal frame $\{X,V,\xi \}$ adapted to the cylinder, where $\xi$ is the unit normal vector field and V is the unit vector field tangent to the fibres of the submersion $\pi$. Thus, the Frenet formula along $\tilde{\gamma }$ reads ${\nabla }_{X}X=\kappa \xi$, ${\nabla }_X\xi =-\kappa X+\tau V$ and ${\nabla }_{X}V=-\tau \xi$. Here, ∇ is the Levi-Civita connection of $(N^3,h)$, while $\kappa$ and $\tau$ represent the geodesic curvature and the torsion of $\tilde{\gamma }$, respectively. Clearly, $\kappa =\overline{\kappa }\circ \pi$. We now denote by A the shape operator of the surface with respect to $\xi$. With respect to the frame $\{X,V,\xi \}$, we can compute the mean curvature, the second fundamental form and other relevant terms appearing in Equation (1) (see, e.g., [7]), which are given by

$$\begin{array}{ccc}\hfill & & A\left(X\right)=-\langle {\nabla }_X\xi ,X\rangle X-\langle {\nabla }_X\xi ,V\rangle V=\kappa X-\tau V,\hfill \\ \hfill & & A\left(V\right)=-\langle {\nabla }_V\xi ,X\rangle X-\langle {\nabla }_V\xi ,V\rangle V=-\tau X;\hfill \\ \hfill & & B(X,X)=\langle A\left(X\right),X\rangle =\kappa ,B(X,V)=\langle A\left(X\right),V\rangle =-\tau ,\hfill \\ \hfill & & B(V,X)=\langle A\left(V\right),X\rangle =-\tau ,B(V,V)=\langle A\left(V\right),V\rangle =0,\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}\hfill & & H={\displaystyle \frac{1}{2}}(B(X,X)+B(V,V))={\displaystyle \frac{\kappa }{2}},\hfill \\ \hfill & & A\left(\mathrm{grad}H\right)=A(X({\displaystyle \frac{\kappa }{2}})X+V({\displaystyle \frac{\kappa }{2}})V)=X({\displaystyle \frac{\kappa }{2}})A\left(X\right)={\displaystyle \frac{{\kappa }^{\prime }}{2}}(\kappa X-\tau V);\hfill \\ \hfill & & \Delta H=XX\left(H\right)+VV\left(H\right)-{\nabla }_{X}X\left(H\right)-{\nabla }_{V}V\left(H\right)={\displaystyle \frac{{\kappa }^{\prime \prime }}{2}};\hfill \\ \hfill & & {\left|A\right|}^2={\left(B(X,X)\right)}^2+{\left(B(X,V)\right)}^2+{\left(B(V,X)\right)}^2+{\left(B(V,V)\right)}^2={\kappa }^2+2{\tau }^2.\hfill \end{array}$$

(24)

It is well known that the Hopf cylinder can be identified with an isometric immersion $\phi :\Sigma \to (N^3,h)$ equipped with the induced metric $g={\phi }^{*}h$ and may also be parametrized as $\phi (s,z)=(\gamma \left(s\right),z)\subset (N^3,h)$ (see, e.g., [3,23]).

We are now ready to present the biharmonic equation for conformal immersions of the Hopf cylinders associated with the Riemannian submersion from $(N^3,h)$.

Proposition 4.

A conformal immersion of the Hopf cylinder $\phi :(\Sigma ,\overline{g}=f^{-1}{\phi }^{*}h)\to (N^3,h)$ is biharmonic if and only if

$$\left\{\begin{array}{c}{\kappa }^{\prime \prime }-{\kappa }^3-2\kappa {\tau }^2+\kappa {\mathrm{Ric}}^N(\xi ,\xi )+2{\kappa }^{\prime }X(lnf)\hfill \\ +\kappa \{XX(lnf)+VV(lnf)+{(Xlnf)}^2+{(Vlnf)}^2\}=0,\hfill \\ 3\kappa {\kappa }^{\prime }-2\kappa {\mathrm{Ric}}^N(\xi ,X)+2{\kappa }^{2}X(lnf)-2\kappa \tau V(lnf)=0,\hfill \\ {\kappa }^{\prime }\tau +\kappa {\mathrm{Ric}}^N(\xi ,V)+\kappa \tau X(lnf)=0.\hfill \end{array}\right.$$

(25)

Proof. 

The result follows by substituting Equations (23) and (24) into the biharmonic Equation (1) and simplifying the resulting expression. □

Recall that we refer to the 3-dimensional Riemannian manifold

$$M^3(m,l)=\left({\mathbb{R}}^3,h={\displaystyle \frac{d{x}^2+d{y}^2}{{[1+m(x^2+y^2)]}^2}}+{\left[dz+{\displaystyle \frac{l}{2}}{\displaystyle \frac{ydx-xdy}{1+m(x^2+y^2)}}\right]}^2\right)$$

as 3-dimensional Bianchi–Cartan–Vranceanu spaces (abbreviated as 3-dimensional BCV spaces). These spaces include the conformally flat spaces ${\mathbb{R}}^3$, $S^3$, $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$, in addition to the model spaces Nil, $\widetilde{SL}(2,\mathbb{R})$ and $SU\left(2\right)$.

A natural orthonormal frame on $M^3(m,l)$ is given by

$$E_1=F_0{\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \frac{ly}{2}}{\displaystyle \frac{\partial }{\partial z}},E_2=F_0{\displaystyle \frac{\partial }{\partial y}}+{\displaystyle \frac{lx}{2}}{\displaystyle \frac{\partial }{\partial z}},E_3={\displaystyle \frac{\partial }{\partial z}},$$

where $F_0=1+m(x^2+y^2)$.

Note that the map $\pi :M^3(m,l)\to \left({\mathbb{R}}^2,h_1={\displaystyle \frac{d{x}^2+d{y}^2}{{[1+m(x^2+y^2)]}^2}}\right)$ defined by $\pi (x,y,z)=(x,y)$ is a Riemannian submersion with totally geodesic fibres, and $E_3={\displaystyle \frac{\partial }{\partial z}}$ is tangent to these fibres. Assume $\gamma :I\to \left({\mathbb{R}}^2,h_1={\displaystyle \frac{d{x}^2+d{y}^2}{{[1+m(x^2+y^2)]}^2}}\right)$ is an immersed regular curve parametrized by arclength, with $\gamma \left(s\right)=\left(x\right(s),y(s\left)\right)$, and let $\tilde{\gamma }:I\to M^3(m,l)$ be its horizontal lift with geodesic curvature $\kappa$ and torsion $\tau$. Hence, the surface $\Sigma ={\bigcup }_{s\in I}{\pi }^{-1}\left(\gamma \left(s\right)\right)$ is a Hopf cylinder in the BCV space $M^3(m,l)$. As mentioned earlier, this Hopf cylinder can be viewed as an isometric immersion $\phi :\Sigma \to M^3(m,l)$ with the induced metric $g={\phi }^{*}h$ and admits the parametrization $\phi (s,z)=(\gamma \left(s\right),z)=(x\left(s\right),y\left(s\right),z)\subset M^3(m,l)$.

For convenience, we define two auxiliary functions:

$$\kappa (s,m,K)=\left\{\begin{array}{c}{\displaystyle \frac{1}{C_1{e}^{2\sqrt{-(4m+K)}s}+D_1{e}^{-2\sqrt{-(4m+K)}s}+\sqrt{4C_1{D}_1+\frac{1}{4m+K}}}},\mathrm{for}4m+K<0,\hfill \\ {\displaystyle \frac{4C_2}{16+C_2^2{(s+D_2)}^2}},\mathrm{for}4m+K=0,\hfill \\ {\displaystyle \frac{1}{C_{3}cos\left(2\sqrt{4m+K}s\right)+D_{3}sin\left(2\sqrt{4m+K}s\right)+\sqrt{\frac{1}{4m+K}+(C_3^2+D_3^2)}}},\mathrm{for}4m+K>0,\hfill \end{array}\right.$$

(26)

and

$$\psi (z,K)=\left\{\begin{array}{c}{a}_{3}sin\left(\sqrt{-K}z\right)+b_{3}cos\left(\sqrt{-K}z\right),\mathrm{for}K<0,\hfill \\ a_{2}z+b_2,\mathrm{for}K=0,\hfill \\ a_1{e}^{\sqrt{K}z}+b_1{e}^{-\sqrt{K}z},\mathrm{for}K>0,\hfill \end{array}\right.$$

(27)

where K, $a_i$, $b_i$, $C_i$ and $D_i$ are constants, $i=1,2,3$.

We now present the classification result of properly biharmonic conformal immersions of the Hopf cylinders $\Sigma$ into $M^3(m,l)$.

Theorem 5.

Suppose $\phi :(\Sigma ,\overline{g}=f^{-1}{\phi }^{*}h)\to M^3(m,l)$ is a proper biharmonic conformal immersion of a Hopf cylinder into a BCV space, with $f^{{\displaystyle \frac{1}{2}}}$ being a nonconstant conformal factor. Then $l=0$, and one of the following two cases holds:

(i)

The Hopf cylinder has a nonzero constant mean curvature $H={\displaystyle \frac{\kappa }{2}}\ne \pm \sqrt{\left|m\right|}$, and the function f is given by

$$f=\left\{\begin{array}{c}{d}_1{e}^{z\sqrt{{\kappa }^2-4m}}+d_2{e}^{-z\sqrt{{\kappa }^2-4m}},\mathrm{for}{\kappa }^2-4m>0,\hfill \\ d_{1}cos\left(z\sqrt{4m-{\kappa }^2}\right)+d_{2}sin\left(z\sqrt{4m-{\kappa }^2}\right),\mathrm{for}{\kappa }^2-4m<0,\hfill \end{array}\right.$$

(28)

where $d_1$, $d_2$ are constants satisfying $d_1^2+d_2^2\ne 0$ to ensure $f>0$.

(ii)

The Hopf cylinder admits a nonconstant mean curvature $H={\displaystyle \frac{\kappa \left(s\right)}{2}}$, and

$$f=\psi \left(z\right){\kappa }^{-3/2}\left(s\right)=\psi (z,K){\kappa }^{-3/2}(s,m,K)>0.$$

Here, $\kappa \left(s\right)=\kappa (s,m,K)$ and $\psi \left(z\right)=\psi (z,K)$ are given by Equations (26) and (27) respectively, with K being a constant.

Proof. 

We adopt the same notations as above. It is clear that if $\kappa =0$, the surface $\Sigma$ is minimal (hence harmonic), and thus any conformal immersion of this surface is also harmonic. For the remainder of the proof, we assume $\kappa \ne 0$.

According to the results in [23], the orthonormal frame $\{X={\displaystyle \frac{x^{\prime }}{F}}{E}_1+{\displaystyle \frac{y^{\prime }}{F}}{E}_2,V=E_3,\xi ={\displaystyle \frac{y^{\prime }}{F}}{E}_1-{\displaystyle \frac{x^{\prime }}{F}}{E}_2\}$ is adapted to the Hopf cylinder $\Sigma ={\cup }_{s\in I}{\pi }^{-1}\left(\gamma \left(s\right)\right)$. Additionally, the following identities hold: ${\mathrm{Ric}}^N(\xi ,\xi )=4m-{\displaystyle \frac{l^2}{2}}$, ${\left(\mathrm{Ric}\left(\xi \right)\right)}^{\top }=0$, and $\tau =-\langle {\nabla }_{X}V,\xi \rangle =-{\displaystyle \frac{l}{2}}$. Substituting these identities into Equation (25), we derive the system of equations below:

$$\left\{\begin{array}{c}{\kappa }^{\prime \prime }-{\kappa }^3-{\displaystyle \frac{\kappa l^2}{2}}+\kappa (4m-{\displaystyle \frac{l^2}{2}})+2{\kappa }^{\prime }X(lnf)\hfill \\ +\kappa \{XX(lnf)+VV(lnf)+{(Xlnf)}^2+{(Vlnf)}^2\}=0,\hfill \\ 3\kappa {\kappa }^{\prime }+2{\kappa }^{2}X(lnf)+\kappa lV(lnf)=0,\hfill \\ l({\kappa }^{\prime }+\kappa X(lnf))=0.\hfill \end{array}\right.$$

(29)

The system (29) is solved by considering two mutually exclusive cases.

Case I: $l=0$.

With $l=0$, Equation (29) becomes

$$\left\{\begin{array}{c}{\kappa }^2{\displaystyle \frac{\partial lnf}{\partial s}}=-{\displaystyle \frac{3}{2}}{\kappa }^{\prime }\left(s\right)\kappa ,\hfill \\ {\kappa }^{\prime \prime }\left(s\right)-{\kappa }^3+4m\kappa +\kappa [{\displaystyle \frac{{\partial }^{2}lnf}{\partial s^2}}+{\displaystyle \frac{{\partial }^{2}lnf}{\partial z^2}}+{\left({\displaystyle \frac{\partial lnf}{\partial s}}\right)}^2+{\left({\displaystyle \frac{\partial lnf}{\partial z}}\right)}^2]+2{\displaystyle \frac{\partial lnf}{\partial s}}{\kappa }^{\prime }\left(s\right)=0.\hfill \end{array}\right.$$

(30)

By Theorem 2.2 in [23], the Hopf cylinder is biharmonic when ${\kappa }^2=4m$. We therefore restrict our attention to the case where ${\kappa }^2\ne 4m$. Clearly, if $\kappa$ is a nonzero constant (with ${\kappa }^2\ne 4m$), the system (30) further reduces to

$${[lnf\left(z\right)]}^{\prime \prime }+{[lnf\left(z\right)]}^{\prime 2}-{\kappa }^2+4m=0,f_s=0,$$

(31)

which admits the solution given in (28).

For nonconstant $\kappa$, straightforward computation based on (30) and subsequent simplification yields

$$\begin{array}{c}lnf=-{\displaystyle \frac{3}{2}}ln\left|\kappa \left(s\right)\right|+ln\psi \left(z\right),\hfill \\ -2\kappa {\kappa }^{\prime \prime }\left(s\right)+3{\kappa }^{\prime 2}\left(s\right)-4{\kappa }^4+16m{\kappa }^2+4{\kappa }^2\left(s\right){\displaystyle \frac{{\psi }^{\prime \prime }\left(z\right)}{\psi \left(z\right)}}=0,\hfill \end{array}$$

(32)

where $\psi \left(z\right)$ is a positive function.

Since the second equation in (32) must hold for all s and z, we can separate variables and conclude that

$$\begin{array}{c}3{\kappa }^{\prime 2}\left(s\right)-2\kappa {\kappa }^{\prime \prime }\left(s\right)=4{\kappa }^2({\kappa }^2-(4m+K))\mathrm{and}{\displaystyle \frac{{\psi }^{\prime \prime }\left(z\right)}{\psi \left(z\right)}}=K,\hfill \end{array}$$

(33)

where K is a constant.

By Proposition 2 in [14], solving the system (33) gives $\kappa \left(s\right)=\kappa (s,m,K)$ (as defined in (26)) and $\psi \left(z\right)=\psi (z,K)$ (as defined in (27)). Consequently,

$$f=\psi \left(z\right){\kappa }^{-3/2}\left(s\right)=\psi (z,K){\kappa }^{-3/2}(s,m,K).$$

Case II: $l\ne 0$.

From the second and third equations of (29), we derive $X(lnf)={\displaystyle \frac{\partial lnf}{\partial s}}=-{\displaystyle \frac{{\kappa }^{\prime }}{\kappa }}$ and $V(lnf)={\displaystyle \frac{\partial lnf}{\partial z}}=-{\displaystyle \frac{{\kappa }^{\prime }}{l}}$. Combined with the nonconstancy of f, these relations imply that $\kappa$ must be a linear function of s; i.e., $\kappa =as+b$, where $a\ne 0$ and b are constants. Substituting $\kappa =as+b$ into the first equation of (29) and simplifying, we obtain $-{(as+b)}^3+(4m-l^2+{\displaystyle \frac{a^2}{l^2}})(as+b)=0$. This identity forces $a=b=0$, which contradicts the assumption that $a\ne 0$.

Summarizing both Case I and Case II, we complete the proof of the theorem. □

Note that when $l=0$, the corresponding BCV space must be $S^2\times \mathbb{R}$, $H^2\times \mathbb{R}$ or ${\mathbb{R}}^3$. By Theorem 5, we immediately derive the following corollary.

Corollary 4.

(i) A proper biharmonic conformal immersion $\phi :(\Sigma ,\overline{g}=f^{-1}{\phi }^{*}h)\to M^3(m,l)$ of a Hopf cylinder into a BCV space, with $f^{{\displaystyle \frac{1}{2}}}$ being a nonconstant conformal factor, exists if and only if the target BCV space is one of $S^2\times \mathbb{R}$, $H^2\times \mathbb{R}$ or ${\mathbb{R}}^3$.

(ii) There exist no proper biharmonic conformal immersions of Hopf cylinders into $S^3$, $Nil$, $\widetilde{SL}(2,\mathbb{R})$ or $SU\left(2\right)$ with a nonconstant conformal factor.

Corollary 5.

(i) Any proper biharmonic conformal immersion of a Hopf cylinder into a BCV space exists only when the target space is one of $S^2\times \mathbb{R}$, $H^2\times \mathbb{R}$, ${\mathbb{R}}^3$ or $SU\left(2\right)$.

(ii) There exists no proper biharmonic conformal immersion of a Hopf cylinder into $S^3$, $\widetilde{SL}(2,\mathbb{R})$ or Nil, regardless of the choice of conformal factor.

Proof. 

By Theorem 2.2 in [23], proper Hopf cylinders in 3-dimensional BCV spaces exist solely in $S^2\times \mathbb{R}$ or $SU\left(2\right)$. Notably, these immersions are isometric (a special subclass of conformal immersions) with the conformal factor set to 1. Combining this result with Corollary 4, we immediately obtain the conclusion of the corollary. □

Example 2.

(1) Let $\gamma :(0,{\displaystyle \frac{\pi }{2\sqrt{2}}})\to S^2=\left({\mathbb{R}}^2,h_1={\displaystyle \frac{d{x}^2+d{y}^2}{{[1+\frac{1}{4}(x^2+y^2)]}^2}}\right)$ be an immersed regular curve parametrized by arclength, with $\gamma \left(s\right)=\left(x\right(s),y(s\left)\right)$ and the geodesic curvature $\kappa ={\displaystyle \frac{1}{\frac{\sqrt{2}}{2}sin\left(2\sqrt{2}s\right)+1}}$. Consider the Hopf cylinder $\phi :(\Sigma ,{\phi }^{*}h)\to S^2\times \mathbb{R}=\left({\mathbb{R}}^3,h={\displaystyle \frac{d{x}^2+d{y}^2}{{[1+\frac{1}{4}(x^2+y^2)]}^2}}+d{z}^2\right)$ defined by $\phi (s,z)=\left(\gamma \right(s),z)$, which has mean curvature $H={\displaystyle \frac{\kappa }{2}}$. Given $\psi =e^z$ and $f=\psi {\kappa }^{-{\displaystyle \frac{3}{2}}}=e^z{\left\{{\displaystyle \frac{\sqrt{2}}{2}}sin\left(2\sqrt{2}s\right)+1\right\}}^{{\displaystyle \frac{3}{2}}}$, the conformal immersion $\phi :(\Sigma ,\overline{g}=f^{-1}{\phi }^{*}h)\to S^2\times \mathbb{R}$ with conformal factor $f^{{\displaystyle \frac{1}{2}}}$ is proper biharmonic.

Indeed, it is straightforward to verify that substituting $m={\displaystyle \frac{1}{4}}$, $K=1$, $C_3=0$ and $D_3={\displaystyle \frac{\sqrt{2}}{2}}$ into Equation (26) yields $\kappa ={\displaystyle \frac{1}{\frac{\sqrt{2}}{2}sin\left(2\sqrt{2}s\right)+1}}<1$. Correspondingly, taking $K=1$, $a_1=1$ and $b_1=0$ in (27) gives the function $\psi =e^z$. By a well-established result in the differential geometry of curves and surfaces, there exists a curve $\gamma :(0,{\displaystyle \frac{\pi }{2\sqrt{2}}})\to S^2$ with $\gamma \left(s\right)=\left(x\right(s),y(s\left)\right)$ whose geodesic curvature is exactly $\kappa ={\displaystyle \frac{1}{\frac{\sqrt{2}}{2}sin\left(2\sqrt{2}s\right)+1}}$. This implies the Hopf cylinder $\Sigma ={\bigcup }_{s\in I}{\pi }^{-1}\left(\gamma \left(s\right)\right)$ in $S^2\times \mathbb{R}$ has mean curvature $H={\displaystyle \frac{\kappa }{2}}$ and can be identified with the isometric immersion $\phi :(\Sigma ,g={\phi }^{*}h)\to \left({\mathbb{R}}^3,{\displaystyle \frac{d{x}^2+d{y}^2}{{[1+\frac{1}{4}(x^2+y^2)]}^2}}+d{z}^2\right)$ via $\phi (s,z)=(\gamma \left(s\right),z)=(x\left(s\right),y\left(s\right),z)\subset S^2\times \mathbb{R}$. Thus, by Theorem 5, we obtain statement (1).

Analogously, we have the following example:

(2) For $f={(1+s^2)}^{{\displaystyle \frac{3}{2}}}z$ with $z>0$, the conformal immersion of the Hopf cylinder into ${\mathbb{R}}^2\times {\mathbb{R}}^{+}$, i.e., $\phi :(\Sigma ,f^{-1}(d{s}^2+d{z}^2))\to ({\mathbb{R}}^2\times {\mathbb{R}}^{+},h=d{x}^2+d{y}^2+d{z}^2)$ defined by $\phi (s,z)=\left(ln4(\sqrt{1+s^2}+s),\sqrt{1+s^2},z\right)$, is a proper biharmonic map. Note that the associated surface has mean curvature $H={\displaystyle \frac{1}{2(1+s^2)}}$.

Remark 4.

Note that biharmonic conformal immersions of cylinders into ${\mathbb{R}}^3$ and of constant mean curvature Hopf cylinders into $S^2\times \mathbb{R}$ have been characterized in [14] and [7], respectively. Our Theorem 5 recovers these existing results.

4. Conclusions

This work focuses on biharmonic conformal immersions of surfaces, particularly Hopf cylinders and 2-spheres, into conformally flat 3-spaces and obtains a series of geometric characterizations and classification results that supplement existing studies in this field.

We first characterize biharmonic conformal immersions of totally umbilical surfaces into general 3-manifolds and prove the rigidity result that such immersions into nonpositively curved 3-manifolds are conformally minimal. We further construct an infinite family of proper biharmonic conformal immersions from 2-spheres to conformal 3-spheres with nonconstant conformal factors and reproduce the classical biharmonic isometric immersion via an alternative method. Additionally, we derive the biharmonic equation for conformal immersions of Hopf cylinders associated with a Riemannian submersion and classify such immersions in standard BCV 3-spaces.

These results enrich the collection of explicit proper biharmonic maps and carry direct theoretical significance for conformal geometry and biharmonic map theory.

Future work will extend the results to higher-dimensional conformally flat manifolds and explore connections between such immersions and other geometric variational problems.