On closed surfaces with nonnegative curvature in the spectral sense
Kai Xu · arXiv (Cornell University) · 2022
We study closed orientable surfaces satisfying the spectral condition $λ_1(-Δ+βK)\geqλ\geq0$, where $β$ 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 $β$. Associated to a positive super-solution $Δφ\leqβKφ$, the conformal metric $φ^{2/β}g$ has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition.