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.

中文翻译：

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Perelman 在圆锥奇点流形上的 $W$-泛函

在本文中，我们在具有孤立圆锥奇点的流形上发展了 Perelman 的 $W$-泛函理论。特别地，我们证明了在具有孤立圆锥奇点的流形上的某个加权 Sobolev 空间上的 $W$-泛函的下界是有限的，并且如果标量曲率满足奇点附近的某些条件，则存在极小值。我们还获得了奇点附近的极小值的渐近阶。