Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index

Pierre Bérard (1) , Ricardo Sa Earp (2)

1 IF - Institut Fourier
2 Departamento de Matematica

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.

Mots clés: stability; Index; Minimal hypersurfaces

Domaines: Géométrie différentielle [math.DG]

Pierre Bérard, Ricardo Sa Earp. Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index. Bollettino dell'Unione Matematica Italiana, 2016, 9 (3), pp.341--468. ⟨hal-00315294v3⟩