2-环面的负质量定理
A negative mass theorem for the 2-Torus
摘要：
对于度量为g的闭曲面M，点p处的罗宾质量m（p）是去除对数奇异性后p=q处的格林函数G（p，q）的值。拉普拉斯质量是罗宾质量的平均值，减去相同面积的圆形球体的罗宾质量值。拉普拉斯质量是一个谱不变量，它是渐近平坦流形的ADM质量的自然模拟。我们证明了，如果M是一个环面，那么在g的共形类上的拉普拉斯质量的最小值是负的。它是由一个（光滑）度量，其中一个得到了一个尖锐的对数Hardy-Littlewood-Sobolev不等式和Onofri型不等式。如果共形类中的平坦度量足够长和薄，则拉普拉斯质量的极小元是非平坦的。
英文摘要：
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-mass is a spectral invariant which is a natural analog of the ADM mass for asymptotically flat manifolds. We show that if M is a torus, then the minimum value of the Laplacian-mass on the conformal class of g is negative. It is attained by a (smooth) metric for which one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and Onofri-type inequality. If the flat metric in the conformal class is sufficiently long and thin, then the minimizer for the Laplacian-mass is non-flat.
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