Let $M$ be a complete, noncompact constant mean curvature hypersurface of finite index in $\mathbb{R}^{n+1}$ . We show that if either $M$ has zero volume entropy, or zero total curvature entropy and $n \leq 5$, or has bounded curvature and is properly embedded, then $M$ is minimal. We obtain similar results in more general ambient manifolds. Moreover the article contains some results of independent interest, about the volume entropy and the bottom of the essential spectrum.

A smooth closed curve of $mathbb{R}^4$ bounds a minimal disk. This disk may develop singularities. Locally, near a singularity, the minimal surface is a cone generated by a curve- link or knot - that lies in a small hypersphere centered around the singularity. We will address here two questions on the topological nature of these knots: In $mathbb{R}^3$ branch points are always unstable, but not anymore in $mathbb{R}^4$ : what are the knots associated to area-minimizing singularities (more specifically are they identical (up to isotopy) to knots associated to singularities of algebraic curves which are special minimal surfaces)? Are there knots that can never be associated to a minimal singularity? Marc Soret estará en Granada desde el 22 al 25 de abril.