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

TL;DR

This paper develops a short-time existence theory for complete Ricci flows originating from rotationally symmetric metrics on Euclidean space, allowing for unbounded curvature and cone-like singularities, expanding the understanding of Ricci flow behavior.

Contribution

It introduces a novel existence theory for Ricci flows with rotational symmetry on b^{n+1} without requiring bounded curvature, including flows with cone-like singularities.

Findings

Constructed Ricci flow solutions with cone-like singularities

Established existence without bounded curvature assumptions

Demonstrated flows originating from noncompact, rotationally symmetric metrics

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.