Biharmonic hypersurfaces in a sphere

Abstract

In this short paper we will survey some recent developments in the geometric theory of biharmonic submanifolds, with an emphasis on the newly discovered Liouville type theorems and applications of known Liouville type theorems in the research of the nonexistence of biharmonic submanifolds. A new Liouville type theorem for superharmonic functions on complete manifolds is proved and its applications in a kind of nonexistence of biharmonic hypersurfaces in a sphere is provided.

Nonexistence of proper p-biharmonic maps and Liouville type theorems I: case of $$p\ge 2$$

“…Han and Feng [10] proved that p-biharmonic maps from a compact (oriented) manifold into a manifold with nonpositive curvature must be harmonic. In noncompact case nonexistence results of proper isometric p-biharmonic maps were proved in [2][9][10] [11][14] [16] . In [11] Han and Zhang proved the following result.…”

Biharmonic and biconservative hypersurfaces in space forms

“…The inequality in the theorem is verified when one imposes natural geometric conditions on |A| 2 or the scalar curvature of M . Still with additional hypotheses of an analytic nature (for example, using new Liouville type theorems for superharmonic functions on complete non-compact manifolds, and asking that a certain function on M built by using f to be of class L p , or asking that f −1 ∈ L p (M ), for some p ∈ (0, ∞)), the second conjecture was proved right by S. Maeta in [53] and Y. Luo and S. Maeta in [52].…”

Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$

“…Our result can be stated as Theorem 3.3. A rotation hypersurface in S m × R defined in (19) is biharmonic if its mean curvature function H solves the equations…”