Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volume

Mehdi Eddaoudi, Alexandre Girouard

TL;DR

The article establishes an effective upper bound for the volume-normalized second nonzero Laplacian eigenvalue $\overline{\lambda}_2(M,g)$ on closed Riemannian manifolds in terms of the conformal volume $V_c(n,M,C)$ of the conformal class. The principal bound is $\overline{\lambda}_2(M,g) < 2^{2/m} m V_c(n,M,C)^{2/m}$, derived by combining Li–Yau style conformal volume arguments with a folding technique that constructs suitable trial functions orthogonal to the first eigenfunction. The authors provide corollaries and explicit bounds for a wide class of manifolds, including spheres, projective spaces, complex and quaternionic projective spaces, tori, and the Klein bottle, highlighting the practical impact of conformal geometry in spectral estimates. The approach generalizes prior second-eigenvalue results in the conformal category and offers a framework for effective eigenvalue bounds across dimensions. These results connect variational characterizations, conformal geometry, and folding methods to obtain concrete, computable bounds for $\lambda_2$ in diverse geometric settings.

Abstract

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds, thereby generalizing many known results.

Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volume

TL;DR

The article establishes an effective upper bound for the volume-normalized second nonzero Laplacian eigenvalue on closed Riemannian manifolds in terms of the conformal volume of the conformal class. The principal bound is , derived by combining Li–Yau style conformal volume arguments with a folding technique that constructs suitable trial functions orthogonal to the first eigenfunction. The authors provide corollaries and explicit bounds for a wide class of manifolds, including spheres, projective spaces, complex and quaternionic projective spaces, tori, and the Klein bottle, highlighting the practical impact of conformal geometry in spectral estimates. The approach generalizes prior second-eigenvalue results in the conformal category and offers a framework for effective eigenvalue bounds across dimensions. These results connect variational characterizations, conformal geometry, and folding methods to obtain concrete, computable bounds for in diverse geometric settings.

Abstract

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds, thereby generalizing many known results.
Paper Structure (9 sections, 12 theorems, 74 equations, 1 table)

This paper contains 9 sections, 12 theorems, 74 equations, 1 table.

Table of Contents

  1. Introduction
  2. The conformal volume of Li and Yau
  3. The main result and its consequences
  4. Explicit upper bounds for TEXT of various spaces
  5. Plan of the paper
  6. An upper bound on the second conformal eigenvalue
  7. Center of mass arguments
  8. Folding spherical caps
  9. Proof of the main result

Key Result

Theorem 1.1

Let $M$ be a closed $m$-dimensional manifold. For each conformal class $C$ on $M$ and each Riemannian metric $g\in C$ that admits a conformal immersion $\phi:(M,C)\to\mathbb{S}^n\subset\mathbb{R}^{n+1}$, with equality if and only if there exists a minimal immersion $\phi:M\to\mathbb{S}^n$ such that and $\phi^{\star}g_{\mathbb{S}^n}=kg$ for some constant $k>0$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • proof : Proof of Corollary \ref{['Coro:lambda2homo']}
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1: Hersch's center of mass
  • Remark 2.2
  • ...and 8 more