arXiv subjects

Tiarlos Cruz

Publications and source records attributed to Tiarlos Cruz.

13 recordsLinked to original sources

arXiv2026

Band Width Estimates and Rigidity of Manifolds with Negative Curvature

We establish optimal Lipschitz lower bounds for proper smooth functions on three-dimensional Riemannian manifolds with Ricci curvature bounded below by negative constants, yielding a new family of width estimates for Riemannian bands using Gromov's $\mu$-bubble method, together with rigidity statements characterizing the equality case. One of the novelties of our approach lies in its ability to handle higher-genus boundary components, revealing a precise interplay between the width, the area of boundary surfaces, and the underlying topology. Finally, for a complete noncompact three-manifold $M$ with bounded geometry and scalar curvature $R_g\ge -6$, whose $H_2(M,\mathbb{Z})$ contains no spherical or toroidal classes, we prove a sharp lower bound for the boundary area. In the equality case, the manifold is shown to be isometric to an infinite hyperbolic band.

math.DG↗

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-charge inequalities and rigidity of time-symmetric initial data sets

In this paper, we establish new area-charge inequalities for the boundary of time-symmetric Einstein-Maxwell initial data sets, in both compact and noncompact cases, under the dominant energy condition. These inequalities lead to novel rigidity theorems with no analogues in the uncharged setting. In the noncompact case, our result is obtained by applying Gromov's $\mu$-bubble technique in a new geometric context.

gr-qc↗

arXiv2025

Curvature Deformations on Complete Manifolds with Boundary

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary. Building upon these results, we explore further deformation scenarios, including those that increase the mean curvature. Finally, we consider the case of deforming complete manifolds with negative scalar curvature on manifolds with noncompact boundary.

math.DG↗

arXiv2024

Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures

Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $\lambda$ and $\mu$, there exist at least two distinct conformal metrics $g_{\lambda,\mu}=e^{2u_{\mu,\lambda}}g$ and $g^{\lambda,\mu}=e^{2u^{\mu,\lambda}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + \mu$ and $h + \lambda,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{\mu,\lambda}$ when $\mu\downarrow 0$ and $\lambda\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles.

math.DG↗

arXiv2023

On static manifolds satisfying an overdetermined Robin type condition on the boundary

In this work, we consider static manifolds $M$ with nonempty boundary $\partial M$. In this case, we suppose that the potential $V$ also satisfies an overdetermined Robin type condition on $\partial M$. We prove a rigidity theorem for the Euclidean closed unit ball $B^3$ in $\mathbb{R}^3$. More precisely, we give a sharp upper bound for the area of the zero set $Σ=V^{-1}(0)$ of the potential $V$, when $Σ$ is connected and intersects $\partial M$. We also consider the case where $Σ=V^{-1}(0)$ does not intersect $\partial M$.

math.DG↗

arXiv2022

On the $σ_2$-curvature and volume of compact manifolds

In this work we are interested in studying deformations of the $σ_2$-curvature and the volume. For closed manifolds, we relate critical points of the total $σ_2$-curvature functional to the $σ_2$-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky (2001) and Z. Hu and H. Li (2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to $σ_2$-curvature which shows that the volume can be controlled by the $σ_2$-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant $σ_2$-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second order properties of the volume functional we show that there is a variation for which geodesic balls are indeed local minimum in a natural direction.

math.DG↗

arXiv2022

Existence and obstructions for the curvature on compact manifolds with boundary

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic.

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

Critical metrics and curvature of metrics with unit volume or unit area of the boundary

Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. For several cases, depending on the signal of the Yamabe invariant, we give sufficient and necessary condition for a smooth function to be the scalar or mean curvature of a metric with constraint on the volume or area of the boundary.

math.DG↗

arXiv2019

Min-max minimal surfaces, horizons and electrostatic systems

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition for non-electromagnetic matter fields. Moreover, we explore a global version of this inequality, and the rigidity in the case of the equality, using a result proved by Marques and Neves.

math.DG↗

arXiv2019

Prescribing the curvature of Riemannian manifolds with boundary

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on $M$ (resp. metric on $M$ with geodesic boundary). In order to provide analogous results for this problem with $n\geq 3,$ we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on $\partial M$ (resp. on $M$) is a mean curvature of a scalar flat metric on $M$ (resp. scalar curvature of a metric on $M$ and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.

math.DG↗

arXiv2019

On the prescribed $Q$-curvature problem in Riemannian manifolds

We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.

math.DG↗