On the dimension of the space of static potentials on three-manifolds

TL;DR

This paper explores how the number of static potentials relates to the geometric and topological features of three-manifolds, providing classifications and new insights into their structure.

Contribution

It offers a partial classification of boundaryless static manifolds based on the dimension of static potentials and analyzes cases with boundary, extending previous techniques.

Findings

If a compact static manifold with boundary has a static potential with zero set disjoint from the boundary, then the space of static potentials is one-dimensional.

The dimension of static potentials constrains the geometric and topological structure of the manifold.

The analysis of zero sets of static potentials is key to understanding the manifold's properties.

Abstract

We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a compact static manifold with boundary admits a static potential whose zero set is disjoint from the boundary, then the space of static potentials is necessarily one-dimensional. These results rely on a careful analysis of the relative positions of the zero sets of linearly independent static potentials - a technique originally introduced by Miao and Tam.