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César Rosales

Publications and source records attributed to César Rosales.

At least 19 recordsLinked to original sources
arXiv2025

On the anisotropic partitioning problem in Euclidean convex domains

We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features about the minimizers and sharp isoperimetric inequalities with respect to convex cones. Besides some geometric measure theory results about the existence and regularity of minimizers, the proofs rely on a second variation formula for the anisotropic area of hypersurfaces with non-empty boundary.

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arXiv2023

Compact anisotropic stable hypersurfaces with free boundary in convex solid cones

We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $Σ$ immersed in $\mathcal{C}$ with free boundary in $\partial\mathcal{C}\setminus\{0\}$ and minimizing, up to second order, an anisotropic area functional under a volume constraint is contained in a Wulff-shape.

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arXiv2022

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

In a round ball $B\subset\mathbb{R}^{n+1}$ 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.

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arXiv2022

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 $M\times\mathbb{R}^k$ 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 $M\times\mathbb{R}^k$ 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.

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arXiv2021

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

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 $\{\mathcal{S}_λ\}_{λ\geq 0}$ with constant mean curvature $λ$. In this work we first prove that the two closed half-spheres of $\mathcal{S}_0$ with boundary $C_0=\{0\}\times\mathbb{S}^1$ 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 $\mathcal{S}_λ$ with $λ>0$ uniquely solves the isoperimetric problem in $(\mathbb{S}^3,g_h)$ for $C^1$ sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

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arXiv2020

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 behaviour 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 $Ω\times\mathbb{R}$ 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 $Ω\times\{t\}$.

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arXiv2019

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

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 capacity of metric balls in $M$ centered at a point $o\in M$. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from $o$. The technique extends to non-compact submanifolds properly immersed in $M$ under certain control on their weighted mean curvature.

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arXiv2018

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

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 $h$-parabolicity or $h$-hyperbolicity of $P$. Second, we employ functions with geometric meaning to describe submanifolds of bounded $h$-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant $h$-minimal submanifolds appearing in the singularity theory of the mean curvature flow.

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arXiv2016

Strongly stable surfaces in sub-Riemannian $3$-space forms

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 surfaces with empty singular set in the sub-Riemannian $3$-space forms by studying those ones containing a vertical line. As a consequence, we are able to find complete strongly stable non-vertical surfaces with empty singular set in the sub-Riemannian hyperbolic $3$-space $\mathbb{M}(-1)$. In relation to the Bernstein problem in $\mathbb{M}(-1)$ we discover strongly stable $C^\infty$ entire minimal graphs in $\mathbb{M}(-1)$ different from vertical planes. These examples are in clear contrast with the situation in the first Heisenberg group, where complete strongly stable surfaces with empty singular set are vertical planes. Finally, we analyze the strong stability of CMC surfaces of class $C^2$ and non-empty singular set in the sub-Riemannian $3$-space forms. When these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set.

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arXiv2015

Existence, characterization and stability of Pansu spheres in sub-Riemannian $3$-space forms

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in particular the description of Pansu spheres in the first Heisenberg group and the sub-Riemannian $3$-sphere. Then, we study variational properties of $\mathcal{S}_λ(p)$ related to the area functional. First, we obtain uniqueness results for the spheres $\mathcal{S}_λ(p)$ as critical points of the area under a volume constraint, thus providing sub-Riemannian counterparts to the theorems of Hopf and Alexandrov for CMC surfaces in Riemannian $3$-space forms. Second, we derive a second variation formula for admissible deformations possibly moving the singular set, and prove that $\mathcal{S}_λ(p)$ is a second order minimum of the area for those preserving volume. We finally give some applications of our results to the isoperimetric problem in sub-Riemannian $3$-space forms.

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arXiv2014

Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $Ω$ be an open half-space or slab in $\mathbb{R}^{n+1}$ 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 $\partialΩ$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partialΩ$ uniquely minimize the weighted perimeter in $Ω$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partialΩ$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $Ω=\mathbb{R}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $ω$ is concave and possibly non-smooth.

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arXiv2013

Free boundary stable hypersurfaces in manifolds with density and rigidity results

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, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on $M$. Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.

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arXiv2013

Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

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 volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.

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arXiv2010

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

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 surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the only ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.

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arXiv2006

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided 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 a Clifford torus. We study area-stationary surfaces with or without a volume constraint in $(\mathbb{S}^3,g_{h})$. By following the ideas and techniques in [RR] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only $C^2$ compact, connected, embedded surfaces in $(\mathbb{S}^3,g_{h})$ with empty singular set and constant mean curvature $H$ such that $H/\sqrt{1+H^2}$ is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in $(\mathbb{S}^3,g_{h})$.

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arXiv2006

Area-stationary surfaces in the Heisenberg group H^1

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine previous Bernstein type theorems in order to describe entire area-stationary graphs over the xy-plane in H^1. A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the known description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.

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arXiv2006

On the isoperimetric problem in Euclidean space with density

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 isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.

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