18 apr 2019 -- 10:00   [open in google calendar]

Scuola Normale Superiore, Aula Bianchi

Abstract.

A cornerstone in the theory of minimal surfaces is Bernstein's theorem, stating that the only entire minimal graphs in Euclidean 3-space are planes. The effort of many mathematicians lead to several generalizations of this statement. The works of Simons, Bombieri-De Giorgi-Giusti, Moser, Lawson-Osserman and Hildebrandt-Jost-Widman are examples of such results: the first two proving that this theorem is true for minimal hypersurfaces of dimension up to 7 and false for higher dimensions; the third proves the theorem for any minimal hypersurface adding a bounded slope condition; for higher codimensionas, L-O have provided counterexamples even under the extra hypothesis on the slope, while the last work cited gave a stronger condition on the slopeto obtain a Bernstein type result. To prove this theorem we develop general techniques to study the geometry of subsets of a complete Riemannian manifold that contain no image of non-constant harmonic maps. We use this to study regions in a Grassmannian manifold with this property, since the Gauss map of a minimal submanifold is a harmonic map with image into Grassmanians. With these ideas we obtain Bernstein type results.