Blowing bubbles on the torus

TL;DR

This paper investigates the spectral properties of the Laplacian on skinny tori, showing how conformal metrics can significantly alter the trace of the inverse Laplacian, contrasting with known invariants like the log-determinant.

Contribution

It demonstrates that conformal deformations can make the trace of the inverse Laplacian on a torus approach that of a sphere, revealing new geometric spectral behaviors.

Findings

Conformal metrics can reduce the trace close to that of a sphere.

Bubbled tori can have trace less than the sphere.

Contrast with log-determinant extremization on tori.

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.