Ivan Miranda de Almeida (IMPA, Brazil) – On the existence of non-compact CMC hypersurfaces with finite index

Let $X$ be a six-dimensional Riemannian manifold with nonnegative sectional curvature that is a Riemannian product of a closed manifold with an Euclidean factor. We prove that every complete, finite index, non-minimal CMC hypersurface immersed in $X$ is compact. This answers affirmatively a question of do Carmo for this class of ambient Riemannian spaces, extending known lower dimensional results.
As a consequence, we complete the classification of two-sided, complete weakly stable CMC hypersurfaces immersed in the space forms of positive curvature in dimension six.
We also show that a complete, finite index CMC hypersurface immersed in the hyperbolic space $\mathbb{H}^6$ with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.