Ricci flow starting from an embedded closed convex surface in $\mathbb{R}^3$

TL;DR

This paper proves the existence and uniqueness of Ricci flow solutions starting from embedded closed convex surfaces in three-dimensional space, with smooth flows converging to the initial convex surface's metric.

Contribution

It establishes the existence and uniqueness of Ricci flows originating from convex surfaces in , with convergence of metrics in the intrinsic sense.

Findings

Existence of Ricci flow from convex surfaces

Uniqueness of the Ricci flow solution

Smooth Ricci flows converge to the initial surface metric

Abstract

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.