##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry

## Jean Steiner

#### UCSD Graduate Student

## Analogs to the Mass and the Positive Mass Theorem on Spheres

##### Abstract:

We describe two mass-like quantities arising from the Green's function for the Laplacian operator on surfaces. The Robin's mass is obtained by regularizing the logarithmic singularity of the Green's function. We show that the Robin's mass is connected to a spectral invariant. On spheres, we introduce a "geometrical mass", which is, a priori, a smooth function on the sphere. The goemetrical mass is shown to be independent of the point on the sphere, and it is also a spectral invariant. Moreover, a connection to a Sobolev-type inequality reveals that it is minimized at the standard round metric. The definition of the geometrical mass is inspired by the roles played by the Green's function for the conformal Laplacian and the Positive Mass Theorem in the solution to the Yamabe Problem.

Host:

### March 5, 2003

### 1:00 PM

### AP&M 7218

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