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Register a new userStability and the index of biharmonic hypersurfaces in a Riemannian manifold
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Ye-Lin Ou
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In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chens conjecture on biharmonic submanifolds.
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Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reillys inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a geodesic sphere $S(p_0,R_0)$ in $N$, but also its enclosed domain is $C^{1,alpha}$-close to a geodesic ball of constant curvature.
The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (cite{SchoenYau1976}). Recently, Seo (cite{Seo14}) generalized Theorem 1.7 (cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (cite{SchoenYau1976}) and 1.8 (cite{Seo14})
In this paper, we study biharmonic Riemannian submersions. We first derive bitension field of a general Riemannian submersion, we then use it to obtain biharmonic equations for Riemannian submersions with $1$-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. These are used to construct examples of proper biharmonic Riemannian submersions with $1$-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions.
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifolds from minimal submanifolds. We also made some comparison study between biharmonic hypersurfaces of Riemannian space forms and the space-like biharmonic hypersurfaces of pseudo-Riemannian space forms.
In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of cite{OW} and cite{FOR}. We derived the biharmonic equation for hypersurfaces in $S^mtimes mathbb{R}$ and $H^mtimes mathbb{R}$ in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi-parallel for $mge 3$, and some classifications of biharmonic surfaces in $S^2times mathbb{R}$ and $H^2times mathbb{R}$ which are constant angle or belong to certain classes of rotation surfaces.
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