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Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds

Kazumasa Narita

Abstract

For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $λ_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant quantity $λ_{k}(g)\mbox{Vol}(M,g)^{2/n}$ as a functional over all the metrics in a fixed conformal class, we derive a second variation formula for the functional. As a corollary, we prove that if the canonical flat metric on a torus is such that the multiplicity of $λ_{1}$ is two, then the flat metric is not a maximal point of the functional in its conformal class. This is a higher dimensional extension of Karpukhin's very recent work.

Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds

Abstract

For a closed Riemannian manifold of dimension , let be the first positive eigenvalue of the Laplace--Beltrami operator and the volume of . Considering the scale-invariant quantity as a functional over all the metrics in a fixed conformal class, we derive a second variation formula for the functional. As a corollary, we prove that if the canonical flat metric on a torus is such that the multiplicity of is two, then the flat metric is not a maximal point of the functional in its conformal class. This is a higher dimensional extension of Karpukhin's very recent work.
Paper Structure (2 sections, 7 theorems, 48 equations)

This paper contains 2 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

If $a^{2}+b^{2}>1$, then $g_{a,b}$ is not a maximal point of $\overline{\lambda_{1}}\restriction_{[g_{a,b}]}$.

Theorems & Definitions (12)

  • Theorem 1.1: Karpukhin
  • Theorem 1.2: EI-conformal
  • Theorem 1.3: Theorem \ref{['Main-Theorem-Tori']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 2 more