Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inradius collapsed manifolds with a lower Ricci curvature bound

Zhangkai Huang, Takao Yamaguchi

TL;DR

The paper analyzes inradius-collapsed sequences of $n$-dimensional manifolds with boundary under lower Ricci bounds, proving that the boundary limit $C_0$ carries an isometric involution and the manifold limit is the quotient $C_0/f$. It develops a framework in which, when the boundary is non-collapsed, the limit space is a non-collapsed RCD$(K,n-1)$ space, and it bounds the number of boundary components to at most two. The analysis combines Wong's extension technique, tangent-space arguments, and quotient/monodromy considerations to obtain a precise geometric-analytic description of the limit, including both compact and noncompact settings. A suite of examples demonstrates the necessity of the hypotheses and clarifies the scope and limitations of the synthetic Ricci framework in boundary-collapsed scenarios.

Abstract

In this paper, we study a family of $n$-dimensional Riemannian manifolds with boundary having lower bounds on the Ricci curvatures of interior and boundary and on the second fundamental form of boundary. A sequence of manifolds in this family is said to be inradius collapsed if their inradii tend to zero. We prove that the limit space $C_0$ of boundaries of inradius collapsed manifolds admits an isometric involution $f$, and that the limit of the manifolds themselves is isometric to the quotient space $C_0/f$. As an application, we show that the number of boundary components of inradius collapsed manifolds is at most two. Moreover, we prove that the limit space has a lower Ricci curvature bound and an upper dimension bound in a synthetic sense if in addition their boundaries are non-collapsed.

Inradius collapsed manifolds with a lower Ricci curvature bound

TL;DR

The paper analyzes inradius-collapsed sequences of -dimensional manifolds with boundary under lower Ricci bounds, proving that the boundary limit carries an isometric involution and the manifold limit is the quotient . It develops a framework in which, when the boundary is non-collapsed, the limit space is a non-collapsed RCD space, and it bounds the number of boundary components to at most two. The analysis combines Wong's extension technique, tangent-space arguments, and quotient/monodromy considerations to obtain a precise geometric-analytic description of the limit, including both compact and noncompact settings. A suite of examples demonstrates the necessity of the hypotheses and clarifies the scope and limitations of the synthetic Ricci framework in boundary-collapsed scenarios.

Abstract

In this paper, we study a family of -dimensional Riemannian manifolds with boundary having lower bounds on the Ricci curvatures of interior and boundary and on the second fundamental form of boundary. A sequence of manifolds in this family is said to be inradius collapsed if their inradii tend to zero. We prove that the limit space of boundaries of inradius collapsed manifolds admits an isometric involution , and that the limit of the manifolds themselves is isometric to the quotient space . As an application, we show that the number of boundary components of inradius collapsed manifolds is at most two. Moreover, we prove that the limit space has a lower Ricci curvature bound and an upper dimension bound in a synthetic sense if in addition their boundaries are non-collapsed.

Paper Structure

This paper contains 12 sections, 52 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ricci limit spaces
  4. RCD spaces
  5. Manifolds with boundary: gluing and extension
  6. Limit spaces of inradius collapsed manifolds
  7. Metric structure of limit spaces
  8. Tangent space and tangent map
  9. Geometric properties of limit spaces
  10. The RCD$(K,n-1)$ structure of limit spaces
  11. The unbounded diameter case
  12. Examples

Key Result

Theorem 1.3

Assume $\{(M_i,\mathrm{g}_i)\}$ is a sequence of inradius collapsed manifolds in $\mathcal{M}(n,H,K,\lambda,{D})$. Let $(C_0,\mathsf{d})$ and $(X,\mathsf{d}_X)$ be the Gromov-Hausdorff limit spaces of $\{(\partial M_i,\mathsf{d}_{\mathrm{g}_{\partial M_i}})\}$ and $\{(M_i,\mathsf{d}_{\mathrm{g}_i})\

Theorems & Definitions (117)

  • Definition 1.1: Inradius collapsed manifolds
  • Definition 1.2
  • Theorem 1.3: Theorems \ref{['thmfffff4.5']} and \ref{['thm4.10']}
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6: Theorems \ref{['thm4.10']} and \ref{['thm4.30']}
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 107 more