Rigidity Theorems For Hypersurfaces Research Articles

Rigidity theorems for hypersurfaces with constant mean curvature

Let M n be a compact oriented hypersurface of a unit sphere $\mathbb{S}^{n + 1} $ (1) with constant mean curvature H. Given an integer k between 2 and n − 1, we introduce a tensor ⌽ related to H and to the second fundamental form A of M, and show that if |⌽|2 ≤ B H,k and tr(⌽ 3) ≤ C n,k |⌽|3, where B H,k and C n,k are numbers depending only on H, n and k, then either |⌽|2 ≡ 0 or |⌽|2 ≡ B H,k . We characterize all M n with |⌽|2 ≡ B H,k . We also prove that if $$\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}$$ and tr(⌽ 3) ≤ C n,k |⌽|3 then |A|2 is constant and characterize all M n with |A|2 in the interval $$\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] $$ . We also study the behavior of |⌽|2, with the condition additional tr(⌽ 3) ≤ C n,k |⌽|3, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup M |⌽|2 = B H,k and this supremum is attained in M n then M n is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n − k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ⌽ 3 is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n − 1 and $$H \geqslant 1/\sqrt {2n - 1} $$ there is a complete hypersurface M n in $\mathbb{S}^{n + 1} $ (1) with constant mean curvature H such that sup M |⌽|2 = B H,k , and this supremum is attained in M n , and which is not a product of spheres.

Rigidity Theorem for Hypersurfaces in a Unit Sphere

By investigating hypersurfaces M n in the unit sphere S n+1(1) with H k = 0 and with two distinct principal curvatures, we give a characterization of torus the . We extend recent results of Perdomo [9], Wang [10] and Otsuki [8].

RIGIDITY OF HYPERSURFACES IN A EUCLIDEAN SPHERE

AbstractThis paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

A rigidity theorem for hypersurfaces with positive Möbius Ricci curvature in $S^{n+1}$

Let $M^{n}(n\geq 3)$ be an immersed hypersurface without umbilic points in the $(n+1)$-dimensional unit sphere $S^{n+1}$. Then $M^{n}$ is associated with a so-called Möbius form $\Phi$ and a Möbius metric $g$ which are invariants of $M^{n}$ under the Möbius transformation group of $S^{n+1}$. In this paper, we show that if $\Phi$ is identically zero and the Ricci curvature $Ric_{g}$ is pinched: $(n-1)(n-2)/n^{2}\leq Ric_{g}\leq$ $(n^{2}-2n+5)(n-2)/[n^{2}(n-1)]$, then it must be the case that $n=2p$ and $M^{n}$ is Möbius equivalent to $S^{p}(1/\sqrt{2})\times S^{p}(1/\sqrt{2})$.

A rigidity theorem for hypersurfaces in a sphere

A rigidity theorem for hypersurfaces in a sphere