Stable capillary hypersurfaces and the partitioning problem in balls with radial weights

In a round ball B ⊂ Rn+1 endowed with an O(n+1)-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in B which is stable capillary in weighted sense and symmetric about some line containing the center of B is homeomorphic to a closed n-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in B for the Gaussian weight is a closed n-disk of revolution. For n = 2 we also show that a compact weighted stable capillary surface in B of genus 0 is a closed disk of revolution

Stable and isoperimetric regions in some weighted manifolds with boundary

In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming local convexity of the boundary and certain behavior of the Bakry–Émery– Ricci tensor we deduce rigidity properties for stable sets by using deformations constructed from parallel vector fields tangent to the boundary. As a consequence, we completely classify the stable sets in some Riemannian cylinders Ω × R with product weights. Finally, we also establish uniqueness results showing that any minimizer of the weighted perimeter for fixed weighted volume is bounded by a horizontal slice Ω × {t}.MINECO, Spain MTM2017-84851-C2-1-PJunta de Andalucía European Commission FQM32

Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

We consider a convex solid cone C in R^{n+1} with vertex at the origin and boundary smooth away from 0. Our main result shows that a compact two-sided hypersurface Sigma immersed in C with free boundary away from 0 and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape. The technique of proof also works for a non-smooth convex cone C provided the boundary of Sigma is away from the singular set of the boundary of C.Grant PID2020-118180GB-I00 funded by MCIN/AEI/10.13039/501100011033Junta de Andalucía grant PY20-0016

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

The authors were supported by MINECO grant MTM2017-84851-C2-1-P and Junta de Andaluc´ıa grants A-FQM-441-UGR18 and FQM325.We consider the sub-Riemannian 3- sphere (S-3, gh) obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 (2008), no. 3, 675-708] that (S-3, g(h)) contains a family of spherical surfaces {S-lambda}(lambda >= 0) with constant mean curvature.. In this work, we first prove that the two closed half-spheres of S-0 with boundary C-0 = {0} x S-1 minimize the sub-Riemannian area among compact C-1 surfaces with the same boundary. We also see that the only C-2 solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere S-lambda with lambda > 0 uniquely solves the isoperimetric problem in (S-3, g(h)) for C-1 sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.MINECO grant MTM2017-84851-C2-1-PJunta de Andaluc´ıa grants A-FQM-441-UGR18 and FQM32

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

For a Riemannian manifold M, possibly with boundary, we consider the Riemannian product M×Rk with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with non-empty boundary and contained within certain regions of M×Rk with suitable weights. Our results include half-space and Bernstein-type theorems in weighted cylinders. We also generalize to this setting some classical properties about the confinement of a compact minimal hypersurface to certain regions of Euclidean space according to the position of its boundary. Finally, we show interesting situations where the statements are applied, some of them in relation to the singularities of the mean curvature flow.Grant PID2020-118180GB-I00 funded by MCIN/AEI/10.13039/501100011033MEC-Feder grant MTM2017-84851-C2-1-PJunta de Andalucía grants A-FQM-441-UGR18, PY20-00164, FQM32