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Rotationally symmetric Ricci Flow on $\mathbb{R}^{n+1}$

Ming Hsiao

TL;DR

This work establishes short-time existence of complete Ricci flows starting from rotationally symmetric, noncollapsed initial data on $\\mathbb{R}^{n+1}$ without assuming initial curvature bounds. Central tools are a pseudolocality theorem for rotationally symmetric flows and a priori volume-ratio estimates under curvature decay, enabling an approximation-then-limit scheme. The authors construct a complete RS Ricci flow emanating from a warped-product metric with a cone-like origin, obtaining a solution with $|\\mathrm{Rm}(g(t))|\le\\Lambda/t$ on a time interval, and prove a corollary that rough initial data can be smoothed away from the origin via this flow, with convergence to the initial metric outside the origin. The results advance noncompact Ricci-flow theory in higher dimensions, providing explicit control mechanisms (via $\\Lambda$ and $T$) without global curvature bounds and illustrating the utility of equivariant compactness for flows with singular initial data.

Abstract

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on $\mathbb{R}^{n+1}$ that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.

Rotationally symmetric Ricci Flow on $\mathbb{R}^{n+1}$

TL;DR

This work establishes short-time existence of complete Ricci flows starting from rotationally symmetric, noncollapsed initial data on without assuming initial curvature bounds. Central tools are a pseudolocality theorem for rotationally symmetric flows and a priori volume-ratio estimates under curvature decay, enabling an approximation-then-limit scheme. The authors construct a complete RS Ricci flow emanating from a warped-product metric with a cone-like origin, obtaining a solution with on a time interval, and prove a corollary that rough initial data can be smoothed away from the origin via this flow, with convergence to the initial metric outside the origin. The results advance noncompact Ricci-flow theory in higher dimensions, providing explicit control mechanisms (via and ) without global curvature bounds and illustrating the utility of equivariant compactness for flows with singular initial data.

Abstract

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on that are noncollapsed at infinity, without assuming bounded curvature. As a consequence, we construct a complete Ricci flow solution coming out of a rotationally symmetric metric, which has a cone-like singularity at the origin and no minimal hypersphere centered at the origin, using an approximation method.

Paper Structure

This paper contains 7 sections, 14 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. An elementary property of Rotationally symmetric metric on $\mathbb{R}^{n+1}$
  3. Volume ratio and Pseudolocality
  4. Pseudolocality theorem
  5. A priori estimates for volume ratio
  6. Proof of Main results
  7. Proof of Corollary \ref{['roughinitialdata']}

Key Result

Theorem 1

Let $(\mathbb{R}^{n+1},g=ds^{2}+f(s)^{2}g_{\text{std}})$ be a complete and rotational symmetric manifold with $\liminf_{s\rightarrow\infty}f(s)>0$. Then there exists a complete and RSRF $(\mathbb{R}^{n+1},g(t))_{t\in[0,T_{g}]}$ with $g(0)=g$ and a constant $\Lambda_{g}>0$ such that on $\mathbb{R}^{n+1}\times(0,T_{g}]$. Furthermore, assuming there are some constants $\varepsilon,\delta,\ell>0$ suc

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Proposition 1: Volume Ratio bounds
  • proof
  • Example 1
  • Lemma 1: Pseudolocality theorem
  • proof : Proof of Lemma \ref{['prioriestimate']}
  • Lemma 2
  • ...and 20 more