arXiv subjects

Leandro F. Pessoa

Publications and source records attributed to Leandro F. Pessoa.

10 recordsLinked to original sources

arXiv2026

Conformal Geometry and Spectral Bounds on Manifolds with Boundary

This work investigates upper bounds for the spectrum of the Steklov-type operator on Riemannian manifolds with boundary. We extend the Fraser-Schoen estimate for the first positive Steklov eigenvalue to higher Steklov eigenvalues, in terms of the relative conformal volume and the isoperimetric ratio. Our approach, which draw on Korevaar's method, further developed by Grigor'yan-Netrusov-Yau and Kokarev, can be adapt to derive a Korevaar-type estimate for the conformal Dirichlet-to-Robin operator on the Euclidean ball, showing that its $k$-th eigenvalue is bounded from above by a multiple of $k^{2/n}$, as well as a corresponding bound in terms of the relative conformal volume for proper conformal immersion into the Euclidean ball. We also establish a lower bound for the number of negative eigenvalues of the Steklov-type problem in terms of the relative conformal volume, with applications to the spectrum of the conformal Dirichlet-to-Robin operator and to the Morse index of type-II stationary capillary hypersurfaces.

math.DG↗

arXiv2025

Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant

We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $Σ$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of the boundary $H^{\partial M}$ bounded from below. We also prove a local rigidity result in the case $Σ$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $Σ$ and is isometric to $(-\varepsilon,\varepsilon)\times Σ, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partialΣ$ is totally geodesic.

math.DG↗

arXiv2023

On splitting complete manifolds via infinity harmonic functions

In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In dimension 2, we extend Savin's theorem on Lipschitz infinity harmonic functions in the plane to every surface with non-negative sectional curvature.

math.DG↗

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.

math.DG↗

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 $\mathbb{H}^3.$

math.DG↗

arXiv2020

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of this work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations $F$ on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable $F$-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the "standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.

math.DG↗

arXiv2020

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

In this paper, we study some potential theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold $M$. Our main result shows that the forward completeness of $M$ can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including $Δ^N_\infty u \ge g(u)$. Also, an $\infty$-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of $M$. Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of $Δ^N_\infty u = g(u)$ on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of $M$.

math.AP↗

arXiv2017

Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel.

math.DG↗

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.

math.DG↗

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

math.DG↗