Strongly stable surfaces in sub-Riemannian 3-space forms

Ana Hurtado,César Rosales

doi:10.1016/j.na.2017.01.003

Abstract

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.

Highlights

Let M be a Sasakian sub-Riemannian 3-manifold
From the first variation formulas, see for instance [25, Sect. 4.1], a surface Σ in M with ∂Σ = ∅ which is a critical point of the area A for any variation preserving the associated volume V has constant mean curvature H in the sense of (3.2)
In the minimal case (H = 0) the strong stability is the classical condition that A′′(0) 0 for any variation

Summary

Introduction

This result suggests that the stability condition is much less restrictive in the hyperbolic model M(−1) whenever 0 H2 < 1 Motivated by this fact, the second author conjectured in [33, Re. 6.10] the existence of complete stable non-vertical CMC surfaces in M(−1) having empty singular set and mean curvature H ∈ [0, 1). In Theorem 5.8 we are able to deduce, as an immediate consequence of Corollary 3.6 (iii), that the regular set of any surface Cλ(Γ) within a 3-dimensional space form is strongly stable This implies in particular that, in order to show instability of Cλ(Γ), one needs to use suitable variations moving the singular set.

Preliminaries

Strongly stable surfaces

Strongly stable surfaces with empty singular set

Strong stability of surfaces with non-empty singular set