Prescribed Mean Curvature Min-Max Theory in Some Non-Compact Manifolds

2022/04/15 by Liam Mazurowski, Mazurowski, Liam · 1 citation
Mathematics · #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2204.07493

openalex publication_date 2022/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension 3≤ n+1 ≤ 7 and consider a smooth function h\colon ℝⁿ⁺¹→ ℝ which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on h, there exists a closed hypersurface Σ in ℝⁿ⁺¹ with mean curvature prescribed by h. Second, let (M³,g) be an asymptotically flat 3-manifold and fix a constant c > 0. We show that, under an additional assumption on M, it is possible to find a closed surface Σ of constant mean curvature c in M.

Cited by

• Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

Related

• Min-max theory for constant mean curvature hypersurfaces
• Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds
• Existence of hypersurfaces with prescribed mean curvature I - Generic min-max
• Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds
• Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory