An alternative proof of lower bounds for the first eigenvalue on manifolds

TL;DR

This paper presents an alternative elliptic proof for the optimal lower bound of the first eigenvalue on manifolds, originally established via heat equation methods, providing a different perspective on the problem.

Contribution

It offers a new elliptic proof of a known eigenvalue lower bound, complementing the existing heat equation-based approach.

Findings

Provides an elliptic proof of the eigenvalue bound

Validates the bound using Ni's method

Complements previous heat equation proofs

Abstract

Recently, Andrews and Clutterbuck [AC13] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [AC13]. More precisely, following Ni's method ([Ni13, Section 6]) we give an elliptic proof of this theorem.