On closed surfaces with nonnegative curvature in the spectral sense

Kai Xu

On closed surfaces with nonnegative curvature in the spectral sense

Differential Geometry 2023-03-20 v2

Authors: Kai Xu

Abstract

We study closed orientable surfaces satisfying the spectral condition $\lambda_1(-\Delta+\beta K)\geq\lambda\geq0$ , where $\beta$ is a positive constant and $K$ is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for $\beta$ . Associated to a positive super-solution $\Delta\varphi\leq\beta K\varphi$ , the conformal metric $\varphi^{2/\beta}g$ has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove H\"older precompactness and almost rigidity results concerning the main spectral condition.

Keywords

closed surfaces spectral condition gauss curvature isoperimetric inequalities almost rigidity stable minimal surfaces

Cite

@article{arxiv.2211.11715,
  title  = {On closed surfaces with nonnegative curvature in the spectral sense},
  author = {Kai Xu},
  journal= {arXiv preprint arXiv:2211.11715},
  year   = {2023}
}

Comments

v2 updates: the article is largely re-written. Two main theorems are added. Section 2 is organized in a better way. Some original proofs in section 4 are simplified. The example in appendix B is replaced. Introduction and abstract are modified accordingly. The title is changed in order to avoid formulas. 26 pages. Comments are welcomed

On closed surfaces with nonnegative curvature in the spectral sense

Abstract

Keywords

Cite

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