Blowing bubbles on the torus

Abstract

We consider the regularized trace of the inverse of the Laplacian on a skinny torus. With its flat metric, a skinny torus has large trace, but we show that there are conformally equivalent metrics making the trace close to that of a sphere of the same area. This behavior is in sharp contrast to that of the log-determinant, a well-known spectral invariant which is extremized at the flat metric on any torus. Our examples are bubbled tori, where you take a sphere, discard polar regions, and glue top to bottom. In a addendum, we belatedly notice that our bubbled tori have trace less than the sphere, and outline how to exploit this to get Okikiolu's result that by means of a conformal factor depending only on longitude, any torus can be made to have trace less than the sphere.