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arXiv · 2408.10144

Biharmonic conformal immersions into a 3-dimensional conformally flat space

Ze-Ping Wang·Xue-Yi Chen
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Abstract

Inspired by the work of Ou [12,17], we study biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first give a characterization of biharmonic conformal immersions of totally umbilical surfaces into a generic 3-manifold. As an application, we give a method to produce biharmonic conformal immersions into a conformally flat 3-space. We then use the method to obtain a classification of biharmonic maps in a family of conformal immersions and construct many examples of biharmonic conformal immersions from a 2-sphere into a conformal 3-sphere. Our examples include proper biharmonic conformal immersions of a 2-sphere minus a point into a conformal 3-sphere with nonconstant conformal factor and the biharmonic isometric immersion $S^2(\frac{1}{\sqrt{2}})\to S^3$ which was found in [2]. Finally, we study biharmonic conformal immersions of Hopf cylinders of a Riemannian submersion.

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Ze-Ping Wang, Xue-Yi Chen. 2024-08-19. Biharmonic conformal immersions into a 3-dimensional conformally flat space. https://arxiv.org/abs/2408.10144

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