Searcharxiv⌕ Search

arXiv subjects

Marcos P. Cavalcante

Publications and source records attributed to Marcos P. Cavalcante.

At least 19 recordsLinked to original sources
arXiv2026

First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

In this paper, we obtain geometric upper bounds for the first eigenvalue $λ_1(J)$ of the Jacobi operator for both closed hypersurfaces and compact hypersurfaces with boundary having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on $λ_1(J)$ and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue $σ_1(J)$ and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.

math.DG↗
arXiv2026

Topology of stable free boundary CMC surfaces under lower Ricci curvature bounds

We establish intrinsic area--length--topology inequalities for compact free boundary constant mean curvature (CMC) surfaces in three-manifolds with Ricci curvature bounded from below. Our main result is obtained from a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator, derived via a balancing argument. This yields a quantitative inequality that does not require stability and captures both interior and boundary contributions. As an application, we obtain explicit topological restrictions for stable free boundary CMC surfaces under a natural curvature pinching condition. In particular, in weakly convex domains, stability forces low topological complexity, with genus at most three and a small number of boundary components. These results show that effective topological control persists even in negatively curved settings, where classical rigidity phenomena are no longer available.

math.DG↗
arXiv2026

Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

Let $(Σ^2,ds^2)$ be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form $L=Δ+V-aK$ together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential $W$ and (in the curvature-corrected setting) the geodesic curvature $κ_g$ of $\partialΣ$. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of $Σ$ and the integrals of $V$ and $W$, obtained via a Hersch balancing argument on the capped surface. As a geometric application, we derive sharp topological restrictions for compact two-sided free boundary minimal surfaces of Morse index at most one inside geodesic balls of negatively curved pinched Cartan--Hadamard $3$-manifolds under a mild radius condition. We also prove complementary upper bounds for first eigenvalues in the closed and Robin settings, including rigidity in the curvature-corrected case, and we establish Steklov-type estimates in a coercive regime where the Dirichlet-to-Neumann operator is well defined for all boundary data.

math.DG↗
arXiv2026

Index estimates for harmonic Gauss maps

Let $Σ$ denote a closed surface with constant mean curvature in $\mathbb{G}^3$, a 3-dimensional Lie group equipped with a bi-invariant metric. For such surfaces, there is a harmonic Gauss map which maps values to the unit sphere within the Lie algebra of $\mathbb{G}$. We prove that the energy index of the Gauss map of $Σ$ is bounded below by its topological genus. We also obtain index estimates in the case of complete non compact surfaces.

math.DG↗
arXiv2026

On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $Δu + f(u) = 0$ in domains with prescribed asymptotic boundary, extending previous results by Bonorino and Klaser originally established for hyperbolic spaces. Using a novel comparison technique based on convex hypersurfaces inspired by Choi, Gálvez, and Lozano, we overcome the absence of totally geodesic foliations, which are instrumental in the hyperbolic space. Our results highlight the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary.

math.AP↗
arXiv2025

Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$.

math.DG↗
arXiv2024

Stability of extremal domains for the first eigenvalue of the Laplacian operator

In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in a general compact Riemannian surface, assuming either nonnegative total Gaussian curvature or small volume.

math.DG↗
arXiv2023

Geometric properties of extremal domains for the $p$-Laplacian operator

In this paper, we explore the geometric properties of unbounded extremal domains for the $p$-Laplacian operator in both Euclidean and hyperbolic spaces. Assuming that the nonlinearity grows at least as the nonlinearity of the eigenvalue problem, we prove that these domains exhibit remarkable geometric properties and cannot be arbitrarily wide. In two dimensions, we prove that such domains with connected complements must necessarily be balls. In the hyperbolic space, we highlight the constraints on extremal domains and the geometry of their asymptotic boundaries.

math.AP↗
arXiv2023

Gap phenomena for constant mean curvature surfaces

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $Σ$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $Σ$ is either a sphere or a right circular cylinder. Next, we show that if $Σ$ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either $Σ$ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3-space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions.

math.DG↗
arXiv2023

Index bounds for closed minimal surfaces in 3-manifolds with the Killing property

Let $Σ$ be a closed minimal surface immersed in a Riemannian 3-manifold carrying an orthonormal Killing frame. This class of ambient spaces includes Lie groups with a bi-invariant metric. In this paper, we prove that the sum of the Morse index and the nullity of $Σ$ is bounded from below by a constant times its genus.

math.DG↗
arXiv2018

Min-oo conjecture for fully nonlinear conformally invariant equations

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius $r\in (0,π/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. As a side product, in dimension $2$ our methods provide a new proof to Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Ampère equations.

math.DG↗
arXiv2018

Vanishing theorems for the cohomology groups of free boundary hypersurfaces

In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ whose size of the traceless second fundamental form is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $\mathbb{B}^{2+k}$.

math.DG↗
arXiv2018

Index Estimates for Free Boundary Constant Mean Curvature Surfaces

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.

math.DG↗
arXiv2015

Uniqueness Theorems for fully nonlinear conformal equations on subdomains of the sphere

In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.

math.DG↗
arXiv2014

Halfspace type Theorems for Self-Shrinkers

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $Σ$ contained in one of the closed half-space determined by $P$ is $Σ= P$." Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller. Also, using a similar geometric idea, we obtain that the only complete self-shrinker properly immersed in an closed cylinder $ \overline{B ^{k+1} (R)} \times \mathbb{R}^{n-k}\subset \mathbb R^{n+1}$, for some $k\in \{1, \ldots ,n\}$ and radius $R$, $R \leq \sqrt{2k}$, is the cylinder $\mathbb S ^k (\sqrt{2k}) \times \mathbb{R}^{n-k}$. We also extend the above results for $λ-$hypersurfaces.

math.DG↗
arXiv2014

Some Isoperimetric Inequalities and Eigenvalue Estimates in Weighted Manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.

math.DG↗