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Capillary minimal slicing and scalar curvature rigidity

Dongyeong Ko, Xuan Yao

Abstract

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain.

Capillary minimal slicing and scalar curvature rigidity

Abstract

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds. In particular, in dimension , we prove following comparison and rigidity statement: given a compact Riemannian -manifold with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in , if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then is isometric to the Euclidean domain.
Paper Structure (20 sections, 38 theorems, 141 equations)

This paper contains 20 sections, 38 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the proof
  4. Discussion in higher dimensions
  5. Organization
  6. Acknowledgements
  7. Preliminaries
  8. Set-up of weighted capillary slicing
  9. Capillary functionals and variational formulas
  10. Angle Identities
  11. Local estimates
  12. Winding Number
  13. Non-trivial minimizer
  14. First Slicing
  15. Second Slicing
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $V$ be a Riemannian manifold diffeomorphic to the $n$-ball the boundary of $V$ of which has positive mean curvature $H_{\partial V} >0$, let $\underline{V} \subset \mathbb{R}^{n}$ be a convex domain with smooth boundary and let $f: \partial V \rightarrow \partial \underline{V}$ is a diffeomorphi

Theorems & Definitions (74)

  • Theorem 1.1: gromov2021lecturesscalarcurvature, Section 3.1. II
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • ...and 64 more