Perelman's $\lambda$-functional on manifolds with conical singularities

TL;DR

This paper extends Perelman's $$-functional to manifolds with isolated conical singularities by analyzing the spectral properties of a Schrf6dinger operator and describing eigenfunction asymptotics near singularities.

Contribution

It establishes the discreteness of the spectrum and eigenfunction behavior for Schrf6dinger operators on singular manifolds, extending Perelman's $$-functional theory.

Findings

Discrete eigenvalues with finite multiplicities are proven.

Eigenfunctions exhibit specific asymptotic behavior near singularities.

The $$-functional theory is extended to singular manifolds.

Abstract

In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schr\"odinger operator $-4\Delta+R$ consists of discrete eigenvalues with finite multiplicities, if the scalar curvature $R$ satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectral properties, we extend the theory of the Perelman's $\lambda$-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.