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First eigenvalue characterization of Clifford hypersurfaces and Veronese surface

PeiYi Wu

Abstract

We give an estimate for the first eigenvalue of the Schrödinger operator $L:=-Δ-σ$ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb{S}^{n+m}$, where $σ$ is the square norm of the second fundamental form.

First eigenvalue characterization of Clifford hypersurfaces and Veronese surface

Abstract

We give an estimate for the first eigenvalue of the Schrödinger operator which is defined on the closed minimal submanifold in the unit sphere , where is the square norm of the second fundamental form.
Paper Structure (3 sections, 7 theorems, 48 equations)

This paper contains 3 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Lu's inequality
  3. Proof of the Main Theorem

Key Result

Theorem A

lu2011normal Let Then either $M$ is totally geodesic, or is one of the Clifford hypersurfaces $M_{r,n-r}$ ($1\leq r\leq n$) in $\mathbb{S}^{n+m}, m\geq 1$, or is a Veronese surface in $\mathbb{S}^{2+m}, m \geq 2$.

Theorems & Definitions (15)

  • Theorem A: Lu
  • Remark 1
  • Theorem 1.1: Main Theorem
  • Lemma 2.1
  • Remark 2
  • proof
  • Remark 3
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4: Lu lu2011normal
  • ...and 5 more