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Vicente Palmer

Publications and source records attributed to Vicente Palmer.

At least 19 recordsLinked to original sources
arXiv2025

Criteria for parabolicity and hyperbolicity of conductive Riemannian manifolds

Motivated by the physics of anisotropic conductive materials we consider a linear elliptic operator $Δ_{\mathcal{W}}$ of divergence type on a Riemannian manifold $(M^{n}, g)$. The operator is determined by the metric $g$ and by a given conductivity, which is modeled by a smooth self adjoint tensor field $\mathcal{W}$ of type $(1,1)$. We establish new conditions for a conductive manifold $(M, g, \mathcal{W})$ to be $\mathcal{W}$-parabolic or $\mathcal{W}$-hyperbolic. Here, by definition, a $\mathcal{W}$-hyperbolic manifold (as opposed to a $\mathcal{W}$-parabolic manifold) admits an effective electric current $J$, i.e. a bounded potential function $u$, which is a solution to the $\mathcal{W}$-Laplace equation $Δ_{\mathcal{W}}(u) = 0$ with a finite flux of the current $J = -\mathcal{W}(\nabla u)$ to infinity. We prove a number of intrinsic conditions on $g$ and $\mathcal{W}$, that tell the type ( $\mathcal{W}$-hyperbolic or $\mathcal{W}$-parabolic) of conductive Riemannian manifolds. And we prove similar extrinsic conditions for submanifolds (involving also naturally the second fundamental form), that give the type of the submanifolds when they are endowed with the inherited conductivities from the ambient conductive space. Our results are furthermore illustrated by corresponding families of examples, which emphasize how the present setting and results generalize previous findings concerning the usual Laplacian for Riemannian manifolds (with homogeneous, constant, conductivity) as well as similar recent results for weighted manifolds and submanifolds. We also present novel examples of $\mathcal{W}$-hyperbolic manifolds where the conductivity tensor is 'extracted' from the curvature tensor of the manifold itself, such as e.g. the metric equivalents of the Einstein tensor and the Schouten tensor.

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arXiv2025

Concentration of mean exit times

The mean exit time function defined on the $δ$-tube around any equator $\mathbb{S}^{n-1} \subseteq \mathbb{S}^{n}$ of the sphere $\mathbb{S}^{n}$, ($0<δ<π/2$), goes to infinity with the dimension, so that when we consider a Brownian particle that begins its motion at one equator of the sphere, this particle will remain near this equator for an almost infinite amount of time when the dimension of the sphere goes to infinity. On the other hand, if the Brownian particle begins its motion at the North pole, then this particle will leave quickly, when the dimension of the sphere goes to infinity, any geodesic ball with radius $δ<π/2$, centered at this point. Namely, the mean exit time function defined on the equatorial tubes presents a kind of {\em concentration} phenomenon or {\em fat equator} effect, as it has been described in the book \cite{MS}. Moreover, the same concentration phenomenon occurs when we consider this mean exit time function defined on tubes around closed and minimal hypersurfaces of a compact Riemannian $n$-manifold $M$ with Ricci curvature bounded from below, ${\rm Ric}_{M}\geq (n-1)$. Namely, a Brownian particle that begins its random movement around a closed embedded minimal hypersurface of a compact $n$-manifold $M$ with ${\rm Ric}_{M}\geq (n-1)$ will wanders arbitrarily close to the hypersurface for a time that approaches infinity as the dimension of the ambient manifold does so as well.

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arXiv2024

Fat equator effect and Minimality in immersions and submersions of the Sphere

Inspired by the equatorial concentration of measure phenomenon in the sphere, a result which is deduced from the general, (and intrinsic), concentration of measure in $\mathbb{S}^n(1)$, we describe in this paper an equatorial concentration of measure satisfied by the closed, (compact without boundary), isometric and minimal immersions $x:Σ^m \rightarrow \mathbb{S}^n(1)$, ($m \leq n$), and by the minimal Riemannian submersions $π: Σ^m \rightarrow \mathbb{S}^n(1)$, ($m \geq n$).

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arXiv2021

First Dirichlet eigenvalue and exit time moment spectra comparisons

We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged $L^1$-moment spectra $\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$, and the torsional rigidity $\mathcal{A}_1(B^M_R)$ of a geodesic ball $B^M_R$ in a Riemannian manifold $M^n$ which satisfies that the mean curvatures of the geodesic spheres $S^M_r$ included in it, (up to the boundary $S^M_R$), are controlled by the radial mean curvature of the geodesic spheres $S^ω_r(o_ω)$ with same radius centered at the center $o_ω$ of a rotationally symmetric model space $M^n_ω$. As a consecuence, we prove a first Dirichlet eigenvalue $λ_1(B^M_R)$ comparison theorem and show that equality with the bound $λ_1(B^ω_R(o_ω))$, (where $B^ω_r(o_ω)$ is the geodesic $r$-ball in $M^n_ω$), characterizes the $L^1$-moment spectrum $\{\mathcal{A}_k(B^M_R)\}_{k=1}^\infty$ as the sequence $\{\mathcal{A}_k(B^ω_R)\}_{k=1}^\infty$ and vice-versa.

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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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arXiv2018

Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

We study some potential theoretic properties of homothetic solitons $Σ^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $\mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $Σ^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on $Σ$ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of \cite{CaoLi}.

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arXiv2015

Asymptotically extrinsic tamed submanifolds

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.

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arXiv2015

Mean curvature, volume and properness of isometric immersions

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riemannian manifold $M$ with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.

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arXiv2013

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.

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arXiv2013

Comparison results for capacity

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then ${\rm Cap}(K)\geq (n-1)\,H_0{\rm vol}(\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature, has smooth boundary, and the mean curvature of $\partial K$ is smaller than or equal to $H_0$, we prove the inequality ${\rm Cap}(K)\leq (n-1)\,H_0{\rm vol}(\partial K)$. In both cases we are able to characterize the equality case. Finally, if $K$ is a convex set in Euclidean space $\mathbb{R}^{n+1}$ which admits a supporting sphere of radius $H_0^{-1}$ at any boundary point, then we prove ${\rm Cap}(K)\geq (n-1)\,H_0\mathcal{H}^n(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$.

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arXiv2012

Volume Growth, Number of Ends and the Topology of a Complete Submanifold

Given a complete isometric immersion $ϕ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient manifold is a radially symmetric space, it is shown an inequality between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends which generalizes the classical inequality stated in Anderson's paper "The compactification of a minimal submanifold by the Gauss Map", (Preprint IEHS, 1984), for complete and minimal submanifolds in $\erre^n$. We obtain as a corollary the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in the Hyperbolic space together with Bernstein type results for such submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work due to A. Kasue and K. Sugahara "Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms", (Osaka J. Math. 24,1987).

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arXiv2012

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and Hyperbolic spaces

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in $\erre^n$ and in $\Han$), based in the isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.

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arXiv2010

Comparison of exit moment spectra for extrinsic metric balls

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in $N$. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds $N^n$ themselves.

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arXiv2009

A note on the p-Parabolicity of Submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p bigger or equal than 2.

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