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Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature

Jia-Yong Wu

TL;DR

This work studies complete weighted manifolds under a spectrum-wise lower bound for the infinity-Bakry-Emery Ricci curvature $\mathrm{Ric}_f$, proving three main results: a diameter bound, a global weighted volume bound, and a spectral splitting theorem. The authors formulate a key condition $u\,\mathrm{Ric}_f-\gamma\Delta_f u \ge (n-1)\lambda u$ with $0\le \gamma \le 1$ and $u>0$, and develop mu-bubble and isoperimetric-profile techniques to derive sharp geometric inequalities. The diameter and volume bounds extend prior results of AX17 and CH to the $N=\infty$ setting and connect to Yeung's splitting results, while the splitting theorem for $\mathrm{Ric}_f\ge 0$ in spectrum sense generalizes existing finite-$N$ theory. The results have implications for shrinking gradient Ricci solitons and the broader spectral geometry of weighted manifolds.

Abstract

In this paper, we prove the diameter comparison, the global weighted volume comparison and the splitting theorem in weighted manifolds when the infinity-Bakry-Emery Ricci curvature has a lower bound in the spectrum sense. Our results extend Antonelli-Xu's spectral Bonnet-Myers and Bishop-Gromov theorems, and Antonelli-Pozzetta-Xu's spectral splitting theorem to weighted manifolds. Our results are also some supplements of Chu-Hao's spectral diameter and global volume comparisons, and Yeung's spectral splitting theorem in weighted manifolds.

Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature

TL;DR

This work studies complete weighted manifolds under a spectrum-wise lower bound for the infinity-Bakry-Emery Ricci curvature , proving three main results: a diameter bound, a global weighted volume bound, and a spectral splitting theorem. The authors formulate a key condition with and , and develop mu-bubble and isoperimetric-profile techniques to derive sharp geometric inequalities. The diameter and volume bounds extend prior results of AX17 and CH to the setting and connect to Yeung's splitting results, while the splitting theorem for in spectrum sense generalizes existing finite- theory. The results have implications for shrinking gradient Ricci solitons and the broader spectral geometry of weighted manifolds.

Abstract

In this paper, we prove the diameter comparison, the global weighted volume comparison and the splitting theorem in weighted manifolds when the infinity-Bakry-Emery Ricci curvature has a lower bound in the spectrum sense. Our results extend Antonelli-Xu's spectral Bonnet-Myers and Bishop-Gromov theorems, and Antonelli-Pozzetta-Xu's spectral splitting theorem to weighted manifolds. Our results are also some supplements of Chu-Hao's spectral diameter and global volume comparisons, and Yeung's spectral splitting theorem in weighted manifolds.

Paper Structure

This paper contains 5 sections, 13 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Diameter comparison
  4. Global weighted volume comparison
  5. Weighted spectral splitting theorem

Key Result

Theorem 1.1

Let $(M,g,e^{-f}dv)$ be a complete $n$-dimensional ($n\geqslant3$) weighted manifold and $\gamma$ be a constant such that $0\leqslant \gamma\leqslant 1$. Assume that there exists a positive function $u\in C^{\infty}(M)$ such that for some constant $\lambda>0$. If $0<\inf_Mu\leqslant\sup_Mu<\infty$ and $F=\|f\|_{C^0(M)}<\infty$, then the following results hold: Here $f_{max}:=\sup_{x\in M} f(x)$

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['h inequality']}
  • Lemma 3.2
  • proof : Proof of (1) in Theorem \ref{['result']}
  • Theorem 3.3
  • ...and 17 more