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An improved upper bound for the second eigenvalue on tori

Fan Kang

Abstract

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $λ_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for $λ_2(T_{a,b},g)$ on any flat torus $T_{a, b}$ with $(a, b)\in \mathbb{R}^2$. Our bound improves the general estimate $λ_2(T_{a, b},g)\le 4A_c(T_{a, b}, [g])$ in the case of the torus. As applications, we derive a uniform upper bound $λ_2(T,g)< \frac{16π^2}{\sqrt{3}}$ for any torus $T$ and any metric $g$, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for $λ_2(T_{a,b},g)$ on the specific family of flat tori $T_{a,b}$ with $0\leq a\leq \frac12$ and $\sqrt{1-a^2}\leq b\leq 1.76$.

An improved upper bound for the second eigenvalue on tori

Abstract

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue on a torus , among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for on any flat torus with . Our bound improves the general estimate in the case of the torus. As applications, we derive a uniform upper bound for any torus and any metric , and reduce the Kao-Lai-Osting conjecture to proving an upper bound for on the specific family of flat tori with and .

Paper Structure

This paper contains 5 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of trial functions for $\lambda_2(M,g)$
  4. Dirichlet energy
  5. Proofs of the main results

Key Result

Theorem 1.1

Let $(a,b)\in\mathscr{M}$ and let $g$ be a Riemannian metric on $T_{a,b}$ conformal to the flat metric $g_{a,b}$. Then where $S=\sqrt{(a^2+b^2)(8+a^2+b^2)}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.1
  • Conjecture 1.1: Kao-Lai-Osting KLO17
  • Corollary 1.3
  • Lemma 2.1: LangesenL21, Corollary 5
  • Lemma 2.2
  • proof
  • Lemma 2.3: BryantB15
  • Proposition 2.4: El Soufi-Ilias-RosEIR97, Proposition 3.1
  • ...and 5 more