Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry

Gioacchino Antonelli, Kai Xu

TL;DR

The paper establishes sharp spectral generalizations of the Bishop-Gromov and Bonnet-Myers theorems by coupling a positive function u with a lower spectral bound on the operator −γΔ+Ric. Central to the approach are two new variational tools: an unequally weighted isoperimetric profile I(v) and unequally warped μ-bubbles, which yield sharp volume and diameter bounds, as well as rigidity when equality is achieved. In dimensions 3–5, with Ric≥0 and a spectral biRic condition outside a compact set, the authors prove sharp isoperimetric control at infinity and linear volume growth, linking spectral bounds to asymptotic geometry and minimal-surface techniques. The results have consequences for the structure of manifolds at infinity, provide a universal diameter bound in a subrange of γ, and connect to Mazet’s work on the stable Bernstein problem, illustrating a deep interplay between spectral geometry, isoperimetry, and geometric analysis.

Abstract

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geq3$ satisfies $$ λ_1\left(-\frac{n-1}{n-2}Δ+\mathrm{Ric}\right)\geq n-1, $$ then $\operatorname{vol}(M)\leq\operatorname{vol}(\mathbb S^{n})$, and $π_1(M)$ is finite. The constant $\frac{n-1}{n-2}$ cannot be improved, and if $\mathrm{vol}(M)=\mathrm{vol}(\mathbb S^n)$ holds, then $M\cong \mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped $μ$-bubbles. As an application, in dimensions $3\leq n\leq 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature. Furthermore, the main result of this paper is applied in Mazet's recent solution of the stable Bernstein problem in $\mathbb R^6$.

New spectral Bishop-Gromov and Bonnet-Myers theorems and applications to isoperimetry

TL;DR

The paper establishes sharp spectral generalizations of the Bishop-Gromov and Bonnet-Myers theorems by coupling a positive function u with a lower spectral bound on the operator −γΔ+Ric. Central to the approach are two new variational tools: an unequally weighted isoperimetric profile I(v) and unequally warped μ-bubbles, which yield sharp volume and diameter bounds, as well as rigidity when equality is achieved. In dimensions 3–5, with Ric≥0 and a spectral biRic condition outside a compact set, the authors prove sharp isoperimetric control at infinity and linear volume growth, linking spectral bounds to asymptotic geometry and minimal-surface techniques. The results have consequences for the structure of manifolds at infinity, provide a universal diameter bound in a subrange of γ, and connect to Mazet’s work on the stable Bernstein problem, illustrating a deep interplay between spectral geometry, isoperimetry, and geometric analysis.

Abstract

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold of dimension satisfies then , and is finite. The constant cannot be improved, and if holds, then . A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped -bubbles. As an application, in dimensions , we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature. Furthermore, the main result of this paper is applied in Mazet's recent solution of the stable Bernstein problem in .
Paper Structure (10 sections, 10 theorems, 142 equations)

This paper contains 10 sections, 10 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Strategy of the proof
  4. Organization of the paper
  5. Notations
  6. Acknowledgments
  7. Unequally warped μ-bubbles and diameter bounds
  8. Unequally weighted isoperimetric profile and volume bounds
  9. Isoperimetry under Ric≥0 and spectral biRic≥n-2
  10. The case n≥8

Key Result

Theorem 1

Let $(M^n,g)$ be an $n$-dimensional compact smooth Riemannian manifold with $n\geqslant 3$, and let $0\leqslant \gamma\leqslant\frac{n-1}{n-2}$, $\lambda>0$. We denote by $\mathrm{Ric}(x):=\inf_{v\in T_xM,\,|v|=1}\mathrm{Ric}_x(v,v)$ the smallest eigenvalue of the Ricci tensor. Assume there is a pos Let $\widetilde{M}$ be the universal cover of $M$, endowed with the pull-back metric. Then we have:

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 1
  • Remark 1: Smoothing the eigenfunctions
  • Theorem 2: Mazet
  • Theorem 3
  • Lemma 1
  • proof : Proof $(n\leqslant7)$
  • proof : Proof of \ref{['2mainthm']} of \ref{['cor:CLMSArbitraryDimension']}
  • Remark 2: No universal diameter bound
  • Lemma 2
  • ...and 20 more