论文标题
Perelman的功能在具有非异形圆锥形奇点的流形上
Perelman's functionals on manifolds with non-isolated conical singularities
论文作者
论文摘要
在本文中，我们通过从Perelman的$λ$功能功能的频谱的角度开始，将Perelman的功能定义在具有非分离圆锥形奇点的流形上。 （我们对非均匀圆锥形奇点的定义包括孤立的圆锥形奇异性。）我们证明，Schrödinger操作员的频谱$-4δ+ r $在具有非均匀圆锥形奇异性的流形上是由有限的多重性组成的离散特征值，规定了有限的多重性，规定了一个较低的cons conse cons conse conses of Cones的标准。这使我们能够在这些单数歧管上定义$λ$功能功能，并在某些加权的Sobolev不平等的帮助下证明，最低$ W $功能功能是有限的。此外，我们获得了征征的某些渐近行为，以及在奇异性附近的$ w $功能的最小化器，以及在分离的圆锥形奇异点附近的特征函数的更精致的最佳部分渐近膨胀。我们还研究了具有更一般奇异性的流形的$-4δ+ r $和Perelman的功能的光谱，即$ r^α$ -HORN奇异性是代数奇异性的原型。
In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's $λ$-functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schrödinger operator $-4Δ+ R$ on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the $λ$-functional on these singular manifolds, and further, to prove that the infimum of $W$-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the $W$-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of $-4Δ+ R$ and Perelman's functionals on manifolds with more general singularities, i.e. the $r^α$-horn singularities which serve as prototypes of algebraic singularities.
