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Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

Keita Kunikawa, Yohei Sakurai

Abstract

Bamler-Zhang have developed geometric analysis on Ricci flow with scalar curvature bound. The aim of this paper is to extend their work to various geometric flows. We generalize some of their results to super Ricci flow whose Muller quantity is non-negative, and obtain Gaussian heat kernel estimates.

Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

Abstract

Bamler-Zhang have developed geometric analysis on Ricci flow with scalar curvature bound. The aim of this paper is to extend their work to various geometric flows. We generalize some of their results to super Ricci flow whose Muller quantity is non-negative, and obtain Gaussian heat kernel estimates.
Paper Structure (13 sections, 25 theorems, 158 equations)

This paper contains 13 sections, 25 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main results
  4. Preliminaries
  5. Notations and basics
  6. Monotonicity of entropy
  7. Reduced geometry
  8. Zhang type gradient estimates
  9. Characterization of super Ricci flow
  10. Distance distortion estimate
  11. Cutoff function
  12. Mean value inequality
  13. Gaussian heat kernel estimate

Key Result

Theorem 1.1

Let $(M,g(t))_{t\in [0,T)}$ be an $n$-dimensional compact super Ricci flow with $T<+\infty$ satisfying $\mathcal{D}(V)\geq 0$ for all vector fields $V$. Then for any $A>0$, there exist positive constants $\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3,\mathcal{C}_4,\mathcal{C}_5,\mathcal{C}_6>0$ dependin

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1: FZ
  • Remark 2.2
  • Proposition 2.3: FZ
  • Theorem 2.4: FZ
  • Proposition 2.5: FZ
  • Proposition 2.6: M1, H, Y1, Y2
  • Proposition 2.7
  • Proposition 2.8: CGT
  • ...and 29 more