正亏格曲面的负质量定理
A negative mass theorem for surfaces of positive genus
摘要：
我们将黎曼曲面（M，g）的“波长平方和”定义为拉普拉斯算子的逆的正则化迹。我们通过缩放和添加常数来归一化，以获得“质量”，该质量是尺度不变的并且在圆形球体处消失。这是广义相对论中ADM质量的闭曲面的一个类比。我们证明，如果M有正亏格，那么在每个共形类上，质量都达到负最小值。对于极小化度量，存在一个尖锐的对数Hardy-Littlewood-Sobolev不等式和一个Moser-Trudinger-Onofri型不等式。
英文摘要：
We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus then on each conformal class, the mass attains a negative minimum. For the minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.
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