Hyperbolic geometric flow (I): short-time existence and nonlinear stability

TL;DR

This paper proves the short-time existence, uniqueness, and nonlinear stability of hyperbolic geometric flow on Euclidean space, deriving related wave equations and discussing connections to Einstein and Ricci flows.

Contribution

It establishes foundational results for hyperbolic geometric flow, including existence, uniqueness, stability, and links to Einstein and Ricci flows.

Findings

Short-time existence and uniqueness of hyperbolic geometric flow

Nonlinear stability on Euclidean space for dimensions > 4

Wave equations for curvatures derived

Abstract

In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave equations satisfied by the curvatures are derived. The relation of hypergeometric flow to the Einstein equation and the Ricci flow is discussed.