Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

2023/02/03 by Bueno, Antonio, López, Rafael
#35B06 #35B50 #35J93 #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.01720

Abstract

Given a C¹ function H defined in the unit sphere \mathbbS², an H-surface M is a surface in the Euclidean space ℝ³ whose mean curvature H_M satisfies H_M(p)=H(N_p), p∈ M, where N is the Gauss map of M. Given a closed simple curve Γ⊂ℝ³ and a function H, in this paper we investigate the geometry of compact H-surfaces spanning Γ in terms of Γ. Under mild assumptions on H, we prove non-existence of closed H-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on H that ensure that if Γ is a circle, then M is a rotational surface. We also establish the existence of estimates of the area of H-surfaces in terms of the height of the surface.

Related

• Helicoidal surfaces of prescribed mean curvature in ℝ³
• Surfaces of prescribed linear Weingarten curvature in ℝ³
• The geometry of constant mean curvature surfaces in ℝ³
• Rotational surfaces of prescribed Gauss curvature in ℝ³
• Rotational surfaces with prescribed curvatures