Half-space Theorems for Minimal Surfaces in Nil$_3$ and Sol$_3$

Ricci curvature, minimal surfaces and sphere theorems

Using an analogue of Myers’ theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two problems for closed manifolds with positive Ricci curvature since they are both false in dimensions greater than three. §

Minimal Surfaces in Euclidean Space

Uniqueness Theorems for Free Boundary Minimal Disks in Space Forms

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. N...

Minimal thinness for subordinate Brownian motion in half-space

— We study minimal thinness in the half-space H := {x = (x̃, xd) : x̃ ∈ Rd−1, xd > 0} for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of H below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy. Résumé. — Nous é...

Space filling minimal surfaces and sphere packings

A space filling minimal surface is defined tu be any embedded minimal surface without boundary with the property (haï the area and genus enclosed by any large spherical