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Gap Theorem on locally conformally flat manifold

Ming Hsiao, Man-Chun Lee

TL;DR

The paper proves a gap theorem for complete non-compact locally conformally flat manifolds with nonnegative Ricci curvature by constructing an immortal Yamabe flow starting from rough data. This flow is shown to become instantaneously bounded in Ricci curvature, allowing reduction to a bounded-curvature regime where classical techniques apply. A suite of tools—heat-kernel estimates along the flow, a localized maximum principle, and barrier constructions from Poisson equations—enables sharp curvature control under a scale-invariant integral decay condition on the scalar curvature. Consequently, if the initial metric has sufficiently fast integral curvature decay, the manifold must be flat, extending prior results in this area and highlighting a robust gap phenomenon without assuming bounded curvature a priori.

Abstract

In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve. Using this, we demonstrate that if the curvature decays quickly enough in an integral sense, then the manifold must be flat. This partially generalizes the results of Chen-Zhu and Ma.

Gap Theorem on locally conformally flat manifold

TL;DR

The paper proves a gap theorem for complete non-compact locally conformally flat manifolds with nonnegative Ricci curvature by constructing an immortal Yamabe flow starting from rough data. This flow is shown to become instantaneously bounded in Ricci curvature, allowing reduction to a bounded-curvature regime where classical techniques apply. A suite of tools—heat-kernel estimates along the flow, a localized maximum principle, and barrier constructions from Poisson equations—enables sharp curvature control under a scale-invariant integral decay condition on the scalar curvature. Consequently, if the initial metric has sufficiently fast integral curvature decay, the manifold must be flat, extending prior results in this area and highlighting a robust gap phenomenon without assuming bounded curvature a priori.

Abstract

In this work, we study a gap phenomenon in locally conformally flat Riemannian manifolds with non-negative Ricci curvature. We construct complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve. Using this, we demonstrate that if the curvature decays quickly enough in an integral sense, then the manifold must be flat. This partially generalizes the results of Chen-Zhu and Ma.

Paper Structure

This paper contains 7 sections, 14 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Evolution equations along Yamabe flows
  3. Heat kernel estimates
  4. A-priori local estimates
  5. Existence of immortal Yamabe flow
  6. Barrier function from Poisson equation
  7. Curvature estimate along Yamabe flow

Key Result

Theorem 1.1

For all $n\geq 5$, there exists $\varepsilon_0(n)>0$ such that the following holds: Suppose $(M,g_0)$ is a complete non-compact manifold which is locally conformally flat and has $\text{\rm Ric}(g_0)\geq 0$. If for all $x\in M$, then $(M,g_0)$ is flat.

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:heatkernelestimate']}
  • Lemma 4.1
  • ...and 19 more