Perelman's $W$-functional on manifolds with conical singularities

TL;DR

This paper extends Perelman's $W$-functional to manifolds with isolated conical singularities, establishing finiteness, existence of minimizers under curvature conditions, and their asymptotic behavior near singularities.

Contribution

It develops the theory of $W$-functional on singular manifolds, proving finiteness, existence, and asymptotics of minimizers in this setting.

Findings

The infimum of $W$-functional is finite on manifolds with conical singularities.

A minimizer of the $W$-functional exists under certain scalar curvature conditions.

The asymptotic order of the minimizer near singularities is characterized.

Abstract

In this paper, we develop the theory of Perelman's $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvature satisfies certain condition near the singularities. We also obtain an asymptotic order for the minimizer near the singularities.