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The geometric size of the fundamental gap

Vincenzo Amato, Dorin Bucur, Ilaria Fragalà

TL;DR

This work delivers a sharp quantitative refinement of the fundamental gap for convex domains by linking the gap size to geometric flatness through a variational, dimension-reducing framework. Central to the approach is a localized, convex Payne–Weinberger-type partition that reduces higher-dimensional problems to one-dimensional Schrödinger problems with measure potentials, underpinned by a new sharp 1D bound and stratified rearrangement tools. The authors prove a rigidity result for the Andrews–Clutterbuck inequality and obtain a quantitative excess term of order $\frac{w_\Omega^6}{D_\Omega^8}$, valid in any dimension, with a parallel Neumann-gap analogue yielding a John-ellipsoid-influenced lower bound. The methods synthesize 1D sharp estimates, concavity/stratified-geometry, and convex-partition analysis, offering a robust path toward geometric-spectral quantitative results on manifolds as well.

Abstract

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, our main result strengthens Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions à la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schrödinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in $\mathbb{R}^N$, thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang.

The geometric size of the fundamental gap

TL;DR

This work delivers a sharp quantitative refinement of the fundamental gap for convex domains by linking the gap size to geometric flatness through a variational, dimension-reducing framework. Central to the approach is a localized, convex Payne–Weinberger-type partition that reduces higher-dimensional problems to one-dimensional Schrödinger problems with measure potentials, underpinned by a new sharp 1D bound and stratified rearrangement tools. The authors prove a rigidity result for the Andrews–Clutterbuck inequality and obtain a quantitative excess term of order , valid in any dimension, with a parallel Neumann-gap analogue yielding a John-ellipsoid-influenced lower bound. The methods synthesize 1D sharp estimates, concavity/stratified-geometry, and convex-partition analysis, offering a robust path toward geometric-spectral quantitative results on manifolds as well.

Abstract

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in . The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, our main result strengthens Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions à la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schrödinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in , thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang.
Paper Structure (9 sections, 23 theorems, 261 equations)

This paper contains 9 sections, 23 theorems, 261 equations.

Table of Contents

  1. Introduction and statement of the results
  2. The sharp one-dimensional lower bound
  3. Weighted equipartitions à la Payne-Weinberger
  4. Local strengthened version and rigidity of Andrews-Clutterbuck inequality
  5. Estimate of the excess: the geometric play of cells in $N$ dimensions
  6. The case $N=2$
  7. The case $N \geq 3$
  8. The Neumann gap
  9. Appendix

Key Result

Theorem 1

Let $N \geq 2$. There exists a dimensional constant $\overline c>0$ such that, for every open bounded convex domain $\Omega$ in $\mathbb{R} ^N$ with diameter $D_\Omega$ and width $w_\Omega$, we have

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • Remark 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 29 more