A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Sharief Deshmukh,Ibrahim Al-Dayel

doi:10.3390/math8020294

Sharief Deshmukh, Ibrahim Al-Dayel

Open Access

https://doi.org/10.3390/math8020294

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Abstract

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere S 2 n + 1 with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere S 2 n + 1 .

Highlights

Let M be a compact minimal hypersurface of the unit sphere Sn+1 with shape operator A
Simons [1] has shown that on a compact minimal hypersurface M of the unit sphere Sn+1 either A = 0, or k Ak2 = n, or k Ak2 ( p) > n for some point p ∈ M, where k Ak is the length of the shape operator
If for every point p in M, the square of the length of the second fundamental form of M is n, it is known that M must be a subset of a Clifford minimal hypersurface r !

Summary

A Note on Minimal Hypersurfaces of an Odd

Department of Mathematics, College of Science, King Saud University, P.O.Box-2455, Riyadh 11451, Saudi Arabia Department of Mathematics and Statistics, College of Science, Imam Muhammad Ibn Saud Islamic Received: 17 January 2020; Accepted: 11 February 2020; Published: 21 February 2020

Introduction

Proof of Theorem 1

Proof of Theorem 2