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Sharp $L^2$ estimates for the drift heat equation on shrinking Ricci solitons

Heather Macbeth

Abstract

We prove an $L^2$ estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.

Sharp $L^2$ estimates for the drift heat equation on shrinking Ricci solitons

Abstract

We prove an estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.
Paper Structure (14 sections, 14 theorems, 78 equations)

This paper contains 14 sections, 14 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Pseudo-Gaussian $L^2$ bounds for the drift heat equation
  4. Application to the heat equation along Ricci flow
  5. Acknowledgements
  6. Pseudo-Gaussian bound from Bakry-Emery hypercontractivity estimates
  7. Examples
  8. Pseudo-Gaussian bound in the critical case
  9. Estimates given bounded initial data
  10. Derivative computation
  11. Proof of \ref{['main-result']}
  12. The heat equation along the Ricci flow
  13. Transferring the estimate to a bound along the Ricci flow
  14. Example: Euclidean space

Key Result

Theorem 1.1

Let $v\in L^2(e^{-f}\operatorname{dVol}_g)$. Then for all $t\geq 0$,

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1: Bakry-Emery BE85
  • Proposition 2.2
  • proof
  • Example 2.3
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • ...and 18 more