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Comparison Theorems and the Intermediate Ricci Curvature Assumption

Yujie Wu

TL;DR

This work studies obstructions and geometric consequences of $m$-intermediate curvature via stable weighted slicings, extending classical Bonnet–Meyers phenomena to higher-order curvature notions. It employs $\mu$-bubble techniques and stability inequalities to derive diameter and in-radius bounds for slicings under positive $C_m$ and free boundary or spectral curvature conditions. Key contributions include a Shen–Ye–style Bonnet–Meyers bound for stable weighted slicings of order $m-1$ with sharp constants, nonexistence/existence results for stable weighted free boundary slicings under boundary convexity, and explicit diameter/in-radius bounds in the spectral setting. Collectively, these results tie topological and geometric obstructions to generalized curvature conditions and extend the spectral-mean-convex framework to free boundary and weighted slicing contexts, offering quantitative control in settings beyond classical Ricci curvature.

Abstract

We explore the notion of m-intermediate Ricci curvature assumption introduced by Brendle-Hirsch-Johne further. If a manifold has non-negative m-intermediate Ricci curvature and stable weighted slicing of order m-1, then the last slice has almost non-negative Ricci curvature in the spectral sense. We prove comparison theorems on the diameter and in-radius bound for stable weighted (respectively free boundary) slicing in such manifolds (respectively with mean convex boundary).

Comparison Theorems and the Intermediate Ricci Curvature Assumption

TL;DR

This work studies obstructions and geometric consequences of -intermediate curvature via stable weighted slicings, extending classical Bonnet–Meyers phenomena to higher-order curvature notions. It employs -bubble techniques and stability inequalities to derive diameter and in-radius bounds for slicings under positive and free boundary or spectral curvature conditions. Key contributions include a Shen–Ye–style Bonnet–Meyers bound for stable weighted slicings of order with sharp constants, nonexistence/existence results for stable weighted free boundary slicings under boundary convexity, and explicit diameter/in-radius bounds in the spectral setting. Collectively, these results tie topological and geometric obstructions to generalized curvature conditions and extend the spectral-mean-convex framework to free boundary and weighted slicing contexts, offering quantitative control in settings beyond classical Ricci curvature.

Abstract

We explore the notion of m-intermediate Ricci curvature assumption introduced by Brendle-Hirsch-Johne further. If a manifold has non-negative m-intermediate Ricci curvature and stable weighted slicing of order m-1, then the last slice has almost non-negative Ricci curvature in the spectral sense. We prove comparison theorems on the diameter and in-radius bound for stable weighted (respectively free boundary) slicing in such manifolds (respectively with mean convex boundary).

Paper Structure

This paper contains 5 sections, 14 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Bonnet-Meyers For Slicing of Order $m-1$
  4. stable weighted free boundary slicing and In-Radius Bound
  5. Spectral Non-negative Ricci

Key Result

Theorem 1.3

Let $(N^n,g)$ be a closed Riemannian manifold, $2\leq m\leq n-1, m^2-2-n(m-2)\geq 0$ and $N$ has $C_m>0$, then $N$ has no stable weighted slicing of order $m$ (such that the last slice is compact). If $n\leq 7$ (in this case we always have $m^2-2-n(m-2)\geq 0$), suppose $(N^n,g)$ is a closed orienta

Theorems & Definitions (32)

  • Definition 1.1: Brendle-Hirsh-Johne, brendle2024generalization
  • Remark 1.2
  • Theorem 1.3: Brendle-Hirsh-Johne, brendle2024generalization
  • Theorem 1.4: Shen-Ye, shen1996stable
  • Theorem 1.5: Bonnet-Meyers Theorem for the $(m-1)$-th Slice
  • Remark 1.6
  • Definition 1.7: $m$-convexity assumption
  • Theorem 1.8: Stable Weighted Free Boundary Slicing
  • Corollary 1.9
  • Theorem 1.10: In-Radius Bound
  • ...and 22 more