Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity

TL;DR

This paper establishes short-time existence of static flow on complete noncompact asymptotically static manifolds and computes asymptotic expansions at conformal infinity, linking evolution equations to static Einstein vacuum solutions.

Contribution

It provides the first analysis of short-time existence for static flow on such manifolds and derives detailed asymptotic expansions at conformal infinity.

Findings

Short-time existence of static flow established.

Asymptotic expansions of metric and potential at conformal infinity computed.

Stationary points correspond to static Einstein vacuum solutions.

Abstract

In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum $(M^n,g,V),$ we also compute the asymptotic expansions of $g$ and $V$ at conformal infinity.