| Résume | In this talk, we will study the behaviour of stable (or δ-stable) complete mini-mal hypersurfaces in Euclidean space and, more generally, in manifolds Mn+1 with non-negative sectional curvature. Are they all totally geodesic for small enough n? What is their topology? We will discuss some new tools discovered in joint works with X. Cabr e, G. Catino, P. Mastrolia and A. Roncoroni. The guideline is that stable minimal hypersurfaces in ambient spaces with Sec ≥ 0 are an example of manifolds with spectral Ricci lower bounds, a class of spaces for which we provide a generalization of Cheeger-Gromoll’s splitting theorem and sharp pointwise gra- dient estimates for the Green kernel. Among the applications, we get a new proofof the Stable Bernstein Theorem in R4 and a characterization of the 3D-catenoid in R4. |
