Rigidity of Complete Minimal Submanifolds in Spheres

Jundong Zhou

doi:10.3389/fphy.2020.571250

Abstract

Let $M$ be an $n$-dimensional complete minimal submanifold in an $(n+p)$-dimensional sphere $S^{n+p}$ and $h$ be the second fundamental form of $M$. In this paper, it is shown that $M$ is totally geodesic if the $L^2$ norm of $|h|$ on any geodesic ball of $M$ has less than quadratic growth and $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Farther, deleting the condition of the $L^2$ norm of $|h|$, we show that $M$ is totally geodesic if $L^n$ norm of $|h|$ on $M$ is less than a fixed constant. Furthermore, we provide a sufficient condition such that a complete stable minimal hypersurface is totally geodesic.

Highlights

Let x, u(x) be a minimal graph in R2 × R, which means that u(x) solves the equation ∇u div = 0. 1 + |∇u|2The celebrated Bernstein theorem states that the complete minimal graphs in R3 are planes
The works of Fleming [9], Almgren [1], and Neto and Wang [16] tell us that the Bernstein theorem is valid for complete minimal graphs in Rn+1 provided that n ≤ 7
Counterexamples to the theorem for n ≥ 8 have been found by Bombieri et al [2] and, later, by Lawson [13]

Summary

INTRODUCTION

Let x, u(x) be a minimal graph in R2 × R, which means that u(x) solves the equation. = 0. Xia and Wang [20] studied complete minimal submanifolds in a hyperbolic space and obtained the following result. [8] For n ≥ 4, let M be an n-dimensional complete immersed minimal submanifold in a hyperbolic space Hn+p such that n and p satisfy Deshmukh [5] studied n-dimensional compact minimal submanifolds in Sn+p with scalar curvature S satisfying the pinching condition S > n(n − 2) and proved that for p ≤ 2 these submanifolds are totally geodesic. We investigate stable minimal hypersurfaces in the unit sphere and obtain a result similar to do Carmo and Peng’s theorem. For n ≥ 2, let M be an n-dimensional complete stable minimal hypersurface in the unit sphere Sn+1. Where Bx0 (R) denotes the geodesic ball of radius R centered at x0 ∈ M, M is totally geodesic

PRELIMINARIES

PROOFS OF THE MAIN THEOREMS

We need the which implies

CONCLUSION