Ye-Lin Ou

Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

Abstract We prove that a totally umbilical biharmonic surface in any 3 -dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston’s 3-dimensional geometries is proper biharmonic if and only if it is a part of S 2 ( 1 / 2 ) in S 3 . We also give complete classifications of constant mean curvature proper biharmonic surfaces in Thurston’s 3 -dimensional geometries and in 3-dimensional Bianchi–Cartan–Vranceanu spaces, and a complete classification of proper biharmonic Hopf cylinders in 3-dimensional Bianchi–Cartan–Vranceanu spaces.

Biharmonic maps from a 2-sphere

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of rotationally symmetric maps between 2-spheres. We also find many examples of proper biharmonic maps defined locally on a 2-sphere. Our results seem to suggest that any biharmonic map S2⟶(Nn,h) is a weakly conformal immersion.

On f-biharmonic maps and f-biharmonic submanifolds

f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic maps. In this paper, we prove that an f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density is a harmonic map; any f-biharmonic function on a compact manifold is constant, and that the inversions about

Biharmonic and f-biharmonic maps from a 2-sphere

S^m

Biharmonic Riemannian submersions

for

Biharmonic hypersurfaces in Riemannian manifolds

m\ge 3

p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps

are proper f-biharmonic conformal diffeomorphisms. We derive f-biharmonic submanifolds equations and prove that a surface in a manifold

Some constructions of biharmonic maps and Chen’s conjecture on biharmonic hypersurfaces

(N^n, h)

Biharmonic maps and morphisms from conformal mappings

is an f-biharmonic surface if and only it can be biharmonically conformally immersed into

On conformal biharmonic immersions

(N^n,h)