Surfaces with parallel mean curvature in \ℂPⁿ\×\ℝ\n and \ℂHⁿ\×\ℝ

2010/11/21 by Dorel Fetcu, Fetcu, Dorel, Harold Rosenberg +1
Mathematics · #53A10 #53C42 #53C55 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1011.4647

openalex publication_date 2010/11/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We consider surfaces with parallel mean curvature vector (pmc surfaces) in\n\ℂPⁿ\×\ℝ and \ℂHⁿ\×\ℝ, and, more\ngenerally, in cosymplectic space forms. We introduce a holomorphic quadratic\ndifferential on such surfaces. This is then used in order to show that the\nanti-invariant pmc 2-spheres of a 5-dimensional non-flat cosymplectic space\nform of product type are actually the embedded rotational spheres\nS_H²\⊂ M²\×\ℝ of Hsiang and Pedrosa, where M²\nis a complete simply-connected surface with constant curvature. When the\nambient space is a cosymplectic space form of product type and its dimension is\ngreater than 5, we prove that an immersed non-minimal non-pseudo-umbilical\nanti-invariant 2-sphere lies in a product space M⁴\×\ℝ,\nwhere M⁴ is a space form. We also provide a reduction of codimension\ntheorem for the pmc surfaces of a non-flat cosymplectic space form.\n

Citations

• Minimal Varieties in Riemannian Manifolds

Related

• Constant Mean Curvature Surfaces in \mathbbM²(c)×ℝ and Finite Total Curvature
• Spheres with parallel mean curvature in \mathbbS² × ℍ²
• Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces
• Surfaces with parallel mean curvature vector in complex space forms
• Surfaces with Parallel Mean Curvature Vector in S²xS² and H²xH²