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Upper bound for the total mean curvature of spin fill-ins

Christian Baer

TL;DR

This work proves Gromov's conjecture on an upper bound for the total mean curvature of the boundary of a spin fill-in under a scalar curvature lower bound, in the case that the boundary mean curvature is nonnegative. The authors derive a dimension-dependent inequality $\int_M H \le C(M) + \sqrt{\frac{n}{n+1}}\,\lambda\,\mathrm{vol}(M)$ for spin fill-ins with $\mathrm{scal}_X \ge -\lambda^2$, where $C(M)$ depends only on the boundary manifold's spin structure. The proof deploys boundary value problems for the Dirac operator, a carefully constructed boundary constant $C(M)$ built from intrinsic Dirac data on $M$, and a Weitzenböck-type estimate with a modified connection to control boundary terms. The results connect scalar curvature, mean curvature, and spin geometry to provide a quantitative, geometry-driven bound with explicit dependence on $\lambda$ and the boundary volume, matching known sharp cases in special geometries.

Abstract

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin and the mean curvature is non-negative.

Upper bound for the total mean curvature of spin fill-ins

TL;DR

This work proves Gromov's conjecture on an upper bound for the total mean curvature of the boundary of a spin fill-in under a scalar curvature lower bound, in the case that the boundary mean curvature is nonnegative. The authors derive a dimension-dependent inequality for spin fill-ins with , where depends only on the boundary manifold's spin structure. The proof deploys boundary value problems for the Dirac operator, a carefully constructed boundary constant built from intrinsic Dirac data on , and a Weitzenböck-type estimate with a modified connection to control boundary terms. The results connect scalar curvature, mean curvature, and spin geometry to provide a quantitative, geometry-driven bound with explicit dependence on and the boundary volume, matching known sharp cases in special geometries.

Abstract

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin and the mean curvature is non-negative.
Paper Structure (7 sections, 6 theorems, 39 equations)

This paper contains 7 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. A boundary value problem
  3. Some preparation
  4. Proof of the theorem
  5. Definition of the constant C(M)
  6. The estimate
  7. Concluding remarks

Key Result

Theorem 1

Let $X$ be a compact Riemannian spin manifold of dimension $n+1\ge2$ with smooth boundary $\partial X=M$. Let $\lambda\ge0$ be such that the scalar curvature of $X$ satisfies $\mathrm{scal}_X\ge -\lambda^2$. Assume that the mean curvature $H$ of the boundary satisfies $H\ge0$. Then we have where $C(M)$ is a constant depending only on the Riemannian spin manifold $M$.

Theorems & Definitions (14)

  • Conjecture : Gromov G*p. 232
  • Theorem
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 4 more