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G. Pacelli Bessa

Publications and source records attributed to G. Pacelli Bessa.

At least 19 recordsLinked to original sources
arXiv2026

Half-space theorems for translating solitons of the r-mean curvature flow

In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the (r-1)-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the (r-1)-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.

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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

Gap theorems for complete self-shrinkers of $r$-mean curvature flows

In this paper, we prove gap results for complete self-shrinkers of the $r$-mean curvature flow involving a modified second fundamental form. These results extend previous results for self-shrinkers of the mean curvature flow due to Cao-Li and Cheng-Peng. To prove our results we show that, under suitable curvature bounds, proper self-shrinkers are parabolic for a certain second-order differential operator which generalizes the drifted Laplacian and, even if is not proper, this differential operator satisfies an Omori-Yau type maximum principle.

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arXiv2023

Mean exit times from submanifolds with bounded mean curvature

We show that submanifolds with infinite mean exit time can not be isometrically and minimally immersed into cylinders, horocylinders, cones, and wedges of some product spaces. Our approach is not based on the weak maximum principle at infinity, and thus it permits us to generalize previous results concerning non-immersibility of stochastically complete submanifolds. We also produce estimates for the complete tower of moments for submanifolds with small mean curvature immersed into cylinders.

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arXiv2022

On the Cylinder Theorem in $M^2\times \mathbb{R}^n $

Consider a surface $M^2$ with Gaussian curvature either $< 0$ or $> 0$. We prove that in $M^2\times \mathbb{R}^n$ cylinders are characterized as the hypersurfaces with both the extrinsic and intrinsic curvatures equal to zero.

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arXiv2022

Half-space theorems for $1$-surfaces of $\mathbb{H}^3$

In this paper we investigate the intersection problem for $1$-surfaces immersed in a complete Riemannian three-manifold $P$ with Ricci curvature bounded from below by $-2$. We first prove a Frankel's type theorem for $1$-surfaces with bounded curvature immersed in $P$ when $\text{\rm Ric}_{P} > -2$. In this setting we also give a criterion for deciding whether a complete $1$-surface is proper. A splitting result is established when the distance between the $1$-surfaces is realized, even if $\text{\rm Ric}_{P} \geq -2$. In the hyperbolic space $\mathbb{H}^3$ we show strong half-space theorems for the classes of complete $1$-surfaces with bounded curvature, parabolic $1$-surfaces, and stochastically complete $H$-surfaces with $H<1$. As a by-product of our techniques a Maximum Principle at Infinity is given for $1$-surfaces in $\mathbb{H}^3.$

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arXiv2021

Stochastic half-space theorems for minimal surfaces and $H$-surfaces of $\mathbb{R}^{3}$

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $M\times \R_+$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $\text{Ric}_M \geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $\R^3$ with bounded curvature if $\sup \vert H_{_M}\vert < H$, or ${\rm dist}(M,N)=0$ when $\sup \vert H_{_M}\vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.

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arXiv2016

Dirichlet spectrum and Green function

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $λ_{1}(Ω)=\lim_{k\to \infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(Ω, μ)$-momentum spectrum of a bounded domain $Ω$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(Ω, μ)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.

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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

Curvature estimates for submanifolds immersed into horoballs and horocylinders

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls. The proofs rely on the Hessian comparison theorem for the Busemann function.

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arXiv2013

Comparison principle, stochastic completeness and half-space theorems

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in $\mathbb{H}^n \times \mathbb{R}^\ell$.

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arXiv2013

Spectral, stochastic and curvature estimates for submanifolds of highly negative curved spaces

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $φ\colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_φ\leq \infty$. Our proof holds for the bounded case $r_φ< \infty$, recovering the known results, as well as for the unbounded case $r_φ=\infty$. In both cases, the fundamental ingredient in these estimates is the integrability over $(0, r_φ)$ of the inverse $I_{m}^{-1}$ of the comparison isoperimetric radius. When $r_φ=\infty$, this condition is guaranteed if $N$ is highly negatively curved.

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arXiv2012

Curvature estimates for properly immersed $ϕ_{h}$-bounded submanifolds

Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature estimates for properly immersed cylindrically bounded submanifolds. In this paper we prove these sectional and mean curvature estimates for a larger class of submanifolds, the properly immersed $ϕ$-bounded submanifolds.

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arXiv2012

Maximum principle for semi-elliptic trace operators and geometric applications

Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem

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arXiv2012

Spectral and stochastic properties of the $f$-Laplacian, solutions of PDE's at infinity, and geometric applications

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectral and stochastic properties are analyzed in detail.

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