Conic manifolds under the Yamabe flow

TL;DR

This paper studies the Yamabe flow on manifolds with conical singularities, establishing short-term existence and describing how the conical tips deform over time using advanced regularity theory.

Contribution

It proves well-posedness of the Yamabe flow on conic manifolds and provides an asymptotic expansion of the metric near singularities.

Findings

Short-time existence of Yamabe flow on conic manifolds.

Asymptotic description of metric deformation near conical tips.

Application of maximal $L^q$-regularity theory to singular geometries.

Abstract

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the $L^q$-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal $L^q$-regularity theory for conically degenerate operators.