On closed surfaces with nonnegative curvature in the spectral sense

2022/11/21 by Xu, Kai
#52A40 #53C21 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.11715

Abstract

We study closed orientable surfaces satisfying the spectral condition λ₁(-Δ+βK)≥λ≥0, 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≤βKφ, the conformal metric φ^2/βg has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition.

Related

• Area and spectrum estimates for stable minimal surfaces
• Complete Surfaces with Ends of Non Positive Curvature
• Metrics of positive scalar curvature and unbounded widths
• Singular limit laminations, Morse index, and positive scalar curvature
• Isoperimetric inequality, finite total Q-curvature and quasiconformal map