  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.

Differential Geometry