Strongly stable surfaces in sub-Riemannian 3-space forms

A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new examples o...

Full description

Saved in:
Bibliographic Details
Published in:Nonlinear analysis 2017-05, Vol.155, p.115-139
Main Authors: Hurtado, Ana, Rosales, César
Format: Article
Language:English
Subjects:
Bernstein problem
Curvature
Differential geometry
Manifolds
Mathematical analysis
Nonlinear equations
Planes
Second variation formula
Stability
Stability analysis
Stability criteria
Sub-Riemannian space forms
Surface stability
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A surface of constant mean curvature (CMC) equal to H in a sub-Riemannian 3-manifold is strongly stable if it minimizes the functional area+2Hvolume up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian 3-manifolds. We also produce new examples of C1 complete CMC surfaces with empty singular set in the sub-Riemannian 3-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic 3-space M(−1). In relation to the Bernstein problem in M(−1) we discover strongly stable C∞ entire minimal graphs in M(−1) different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class C2 and non-empty singular set in the sub-Riemannian 3-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2017.01.003