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Almost rigidity results of Green functions with non-negative Ricci curvature

Shouhei Honda

Abstract

This short note provides a survey on rigidity and almost rigidity results of Green functions in a non-smooth setting. We also make some observation on the Cheeger-Yau inequality on RCD spaces of non-negative Ricci curvature with applications.

Almost rigidity results of Green functions with non-negative Ricci curvature

Abstract

This short note provides a survey on rigidity and almost rigidity results of Green functions in a non-smooth setting. We also make some observation on the Cheeger-Yau inequality on RCD spaces of non-negative Ricci curvature with applications.
Paper Structure (12 sections, 8 theorems, 48 equations)

This paper contains 12 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Green function and smoothed distance function
  3. Results
  4. Proof
  5. Proof of Theorem \ref{['Colding']}
  6. $\mathop{\mathrm{RCD}}\nolimits$ spaces
  7. Definition
  8. Non-parabolic $\mathop{\mathrm{RCD}}\nolimits(0, N)$ space
  9. Proof of Theorem \ref{['thm:smooth almost']}
  10. Related rigidity and almost rigidity results
  11. Rigidity
  12. Almost rigidity for $\mathop{\mathrm{RCD}}\nolimits(0, N)$ spaces

Key Result

theorem 1

Let $(M^n,g)$ be a complete non-parabolic (namely (nonpara) holds) Riemannian manifold of dimension $n \ge 3$ with non-negative Ricci curvature $\mathrm{Ric}^g \ge 0$. Then for any fixed $x \in M^n$:

Theorems & Definitions (16)

  • definition 1: Smoothed distance function
  • theorem 1: Sharp gradient estimate and rigidity, C12
  • theorem 2: Almost rigidity, Part I: HP
  • definition 2: $\mathop{\mathrm{RCD}}\nolimits(K, N)$ space
  • definition 3: Non-collapsed $\mathop{\mathrm{RCD}}\nolimits(K, N)$ space
  • definition 4: Non-parabolicity
  • definition 5: Green/smoothed distance functions
  • theorem 3: Sharp gradient estimate and rigidity
  • proposition 1: Compactness/continuity of Green functions
  • proposition 2: Sharp upper bound and rigidity
  • ...and 6 more