Volume renormalization for singular Yamabe metrics
Volume renormalization for singular Yamabe metrics
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DOI：
10.1090/proc/13530
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发表时间：
2016-05
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通讯作者：
C. Graham
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文献类型：
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作者：
C. Graham
本文对带边界的紧致流形上的Loewner-Nirenberg奇异Yamabe度量在给定共形类中的体积进行了重整化。这推广了通常的庞加莱-爱因斯坦度量的体积重整化。在体积展开中的对数项的系数定义了一个共形不变的能量，推广了一个表面的Willmore能量，该表面的变分导数相对于边界超曲面的变化是奇异Yamabe度量本身光滑性障碍的倍数。这种能量的存在回答了Gover和沃尔德龙提出的一个问题。
This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a conformally invariant energy generalizing the Willmore energy of a surface whose variational derivative with respect to variations of the boundary hypersurface is a multiple of the obstruction to smoothness of the singular Yamabe metric itself. The existence of such an energy answers a question raised by Gover and Waldron.