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Parabolic frequency monotonicity on the conformal Ricci flow

Abimbola Abolarinwa, Shahroud Azami

Abstract

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.

Parabolic frequency monotonicity on the conformal Ricci flow

Abstract

This paper is devoted to the investigation of the monotonicity of parabolic frequency functional under conformal Ricci flow defined on a closed Riemannian manifold of constant scalar curvature and dimension not less than 3. Parabolic frequency functional for solutions of certain linear heat equation coupled with conformal pressure is defined and its monotonicity under the conformal Ricci flow is proved by applying Bakry-Emery Ricci curvature bounds. Some consequences of the monotonicity are also presented.
Paper Structure (7 sections, 10 theorems, 78 equations)

This paper contains 7 sections, 10 theorems, 78 equations.

Table of Contents

  1. Introduction and main results
  2. Frequency functionals
  3. Conformal Ricci flow
  4. Main results
  5. Notation and Preliminaries
  6. Proof of main theorem and its applications
  7. General parabolic equations

Key Result

Theorem 1.1

Let $(M^{m+1},g(t),p(t)),\,\,t\in[0,T)$ be a solution to the conformal Ricci flow CRF2 with $\mathscr{R}ic_f\leq (\frac{k(t)}{2h(t)}+\frac{R_g}{m+1})g$ and $R_g=-m(m+1)$. Furthermore, $Q'(t)=0$ only if $v$ is an eigenfunction of $\mathscr{L}_f$ satisfying $-\mathscr{L}_f v=c(t)v$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 4.1
  • ...and 2 more