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

Vladimir Medvedev

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.

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

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.

Paper Structure

This paper contains 12 sections, 20 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Static manifolds without boundary
  3. Explicit examples and related questions
  4. Some examples
  5. General form and geometric interpretation
  6. Dimension of the space of static potentials
  7. Completeness questions
  8. Static manifolds with boundary
  9. Appendix A
  10. Appendix B
  11. Derivation of the table conditions
  12. Algorithm for determining completeness of the metric

Key Result

Theorem 1.1

Assume $\dim \mathcal{P}\geqslant 2$, and let $f_1, f_2$ be two linearly independent static potentials on $(M,g)$. If the zero sets intersect, i.e., $f_1^{-1}(0) \cap f_2^{-1}(0) \ne \varnothing$, then $(M,g)$ is a space form. Specifically, if $R_g > 0$, then $(M,g)$ is homothetic to the standard sp

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.1
  • Corollary 1.5
  • Theorem 1.6
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.2
  • ...and 28 more