An upper bound for the smallest area of a minimal surface in manifolds of dimension four

2017/02/22 by Nan Wu, Wu, Nan, Zhifei Zhu +1 · 1 citation
Mathematics · #52C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1702.07034

openalex publication_date 2017/02/22 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that for any closed 4-dimensional Riemannian manifold M with trivial first homology group, if the Ricci curvature |Ric|≤3, the diameter diam(M)≤ D and the volume vol(M)>v>0, then the area of a smallest 2-dimensional stationary integral varifold in M is bounded by F(v,D), for some function F that only depends on v and D. Our bound for the area is based on the estimation of the first homological filling function of M.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Regularity of Einstein Manifolds and the Codimension 4 Conjecture

Cited by

• Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds

Related

• Homological Filling and Minimal Varifolds in Four-Dimensional Einstein Manifolds
• Area and spectrum estimates for stable minimal surfaces
• Estimates for Minimal Volume and Minimal Curvature on 4-dimensional compact manifolds
• Minimal hypersurfaces of least area
• Diameter, Area, and Mean curvature