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On fill-ins with scalar curvature bounded from below and an inequality of Hijazi-Montiel-Roldán

Simon Brendle, Raphael Tsiamis, Yipeng Wang

TL;DR

The paper studies spin fill-ins with scalar curvature bounded below by $-n(n-1)$ and links the Hijazi--Montiel--Roldán inequality for the boundary mean curvature to Gromov's hyperspherical radius. It provides an alternative, boundary-value Dirac operator proof of the HMR inequality based on Bär and Bär–Ballmann, and also yields a self-contained derivation via Weitzenböck-type identities. By combining the HMR bound with Baer’s inequality $\lambda \le \frac{n-1}{2}\,\operatorname{Rad}(\Sigma)^{-1}$, the paper obtains the sharp estimate $\inf_{\partial M} H \le (n-1)\sqrt{1+\operatorname{Rad}(\Sigma)^{-2}}$ and discusses the rigidity in the equality case. These results elucidate the interplay between scalar curvature, Dirac spectra, and hyperspherical geometry in spin fill-ins, contributing to Gromov’s conjectural picture.

Abstract

We consider fill-ins of spin manifolds with scalar curvature bounded by $-n(n-1)$. Gromov proposed a conjecture relating the infimum of the mean curvature of such a fill-in to the hyperspherical radius. We observe that the inequality conjectured by Gromov follows by combining an inequality of Hijazi-Montiel-Roldán for the first Dirac eigenvalue with a recent theorem of Bär. Moreover, we give an alternative proof of the Hijazi-Montiel-Roldán inequality based on the work of Bär and Bär-Ballmann.

On fill-ins with scalar curvature bounded from below and an inequality of Hijazi-Montiel-Roldán

TL;DR

The paper studies spin fill-ins with scalar curvature bounded below by and links the Hijazi--Montiel--Roldán inequality for the boundary mean curvature to Gromov's hyperspherical radius. It provides an alternative, boundary-value Dirac operator proof of the HMR inequality based on Bär and Bär–Ballmann, and also yields a self-contained derivation via Weitzenböck-type identities. By combining the HMR bound with Baer’s inequality , the paper obtains the sharp estimate and discusses the rigidity in the equality case. These results elucidate the interplay between scalar curvature, Dirac spectra, and hyperspherical geometry in spin fill-ins, contributing to Gromov’s conjectural picture.

Abstract

We consider fill-ins of spin manifolds with scalar curvature bounded by . Gromov proposed a conjecture relating the infimum of the mean curvature of such a fill-in to the hyperspherical radius. We observe that the inequality conjectured by Gromov follows by combining an inequality of Hijazi-Montiel-Roldán for the first Dirac eigenvalue with a recent theorem of Bär. Moreover, we give an alternative proof of the Hijazi-Montiel-Roldán inequality based on the work of Bär and Bär-Ballmann.
Paper Structure (2 sections, 7 theorems, 29 equations)

This paper contains 2 sections, 7 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. An alternative proof of the Hijazi--Montiel--Roldán inequality

Key Result

Theorem 1

Let $(M^n, g)$ be a compact, connected Riemannian spin manifold of dimension $n\ge 3$ with connected boundary $\Sigma$. If $R\geq -n(n-1)$ holds at each point of $M$, then where $\lambda = \sqrt{\lambda_{\text{\rm min}}(\mathcal{D}^2)}$ and $\mathcal{D}$ denotes the Dirac operator on the boundary.

Theorems & Definitions (12)

  • Theorem 1: Hijazi--Montiel--Roldán
  • Definition
  • Corollary 2
  • Proposition 3
  • Proposition 4: C. Bär Baer, Appendix A
  • proof
  • Proposition 5
  • proof
  • Corollary 6
  • proof
  • ...and 2 more