Cesar Rosales

University of Granada | UGR · Department of Geometry and Topology

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 Bernste...

We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that $(\mathbb{S}^3,g_h)$ contains a family of spherical surfaces $\{\mathcal{S}_\lambda\}_{\lambda\geq 0}$ with constant mean...

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary C2 surfaces with a non-empty singular set of curves. Combined with previous results,...

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric quotient and the weighted capaci...

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we deduce criteria ensuring the $h$-...

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring strong stability of surfaces in Sasakian $3$-manifolds. We also produce new examples of $C^1$ complete CMC surfa...

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda\in\mathbb{R}$ with $\lambda^2+\kappa>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_\lambda(p)$ immersed in $M$ with constant mean curvature $\lambda$. Our construction recovers in...

Let be an open half-space or slab in ℝn+1 endowed with a perturbation of the Gaussian measure of the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular t...

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\partial M$. As a consequence, w...

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volu...

For constant mean curvature surfaces of class C 2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete su...

We prove that any C2 complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group H1 is either a Euclidean plane or congruent to the hyperbolic paraboloid t=xy.

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$. Comment: 32 pages, no figures, added reference missed in version 3

We consider area-stationary surfaces, perhaps with a volume constraint, in the Heisenberg group endowed with its Carnot–Carathéodory distance. By analyzing the first variation of area, we characterize area-stationary surfaces as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic...

One of the most celebrated papers by M. do Carmo is his joint work with C. K. Peng [Bull. Am. Math. Soc., New Ser. 1, 903–906 (1979; Zbl 0442.53013)] on the classification of complete orientable stable minimal surfaces in ℝ 3 . The authors, through a clever use of the second variational formula of the area together with some information on the conf...

We consider the sub-Riemannian metric g h on \({\mathbb{S}}^{3}\) given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of...

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are i...

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍn which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant m...

We prove that the isoperimetric profile of a convex domain Ω with compact closure in a Riemannian manifold (Mn+1 ,g) satisfies a second-order differential inequality that only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of Ω. Regularity properties of the profile and topological consequences on isoperimetric...

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a lower bound on the Ricci curvature of $\Omega$. Regularity properties of the profile and topological consequence...

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of...

We consider the isoperimetric problem of minimizing perimeter for given volume in a strictly convex domain R,+1 and prove that, if is rotationally sy- mmetric about some line, then any solution to this problem must be convex.

We gather some results in relation to the isoperimetric question in the sub-Riemannian Heisenberg group ℍ 1 . We pay special attention to a new contribution obtained in [M. Ritoré and the author, Adv. Math. 219, No. 2, 633–671 (2008; Zbl 1158.53022)], where we prove that the conjectured “bubble sets” of P. Pansu [in: Differential geometry on homoge...