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15 July 2005 A geometrical mass and its extremal properties for metrics on S 2
Jean Steiner
Duke Math. J. 129(1): 63-86 (15 July 2005). DOI: 10.1215/S0012-7094-04-12913-6

Abstract

Green's function for the Laplacian on surfaces is considered, and a mass-like quantity is derived from a regularization of Green's function. A heuristic argument, inspired by the role of the positive mass theorem in the solution to the Yamabe problem, gives rise to a geometrical mass that is a smooth function on a compact surface without boundary. The geometrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a sharp Sobolev-type inequality reveals that it is actually minimized at the standard round metric. The behavior of the geometrical mass on the sphere is markedly different from that on other surfaces.

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Jean Steiner. "A geometrical mass and its extremal properties for metrics on S 2 ." Duke Math. J. 129 (1) 63 - 86, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12913-6

Information

Published: 15 July 2005
First available in Project Euclid: 15 July 2005

zbMATH: 1144.53055
MathSciNet: MR2153456
Digital Object Identifier: 10.1215/S0012-7094-04-12913-6

Subjects:
Primary: 35J60
Secondary: 53A30 , 58J50 , 58J52

Rights: Copyright © 2005 Duke University Press

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Vol.129 • No. 1 • 15 July 2005
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