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A dual Yamabe flow and related integral flows

Jingang Xiong

Abstract

We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.

A dual Yamabe flow and related integral flows

Abstract

We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.
Paper Structure (10 sections, 29 theorems, 227 equations)

This paper contains 10 sections, 29 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. A general framework and main theorems
  4. Separable solutions
  5. Existence and regularity
  6. Bounds at large time
  7. The critical regime
  8. Long time existence and concentration compactness
  9. Global bound via the moving spheres method
  10. Convergence

Key Result

Theorem 1.1

Let $u$ be a solution of eq:IP with $u^m \in C^1([0,T^*); C^0(M))$ for some $T^*>0$ representing the maximum existence time. Then we have:

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop:critial']}
  • Lemma 3.1
  • ...and 47 more