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José M. Espinar

Publications and source records attributed to José M. Espinar.

At least 19 recordsLinked to original sources
arXiv2026

Frobenius conjugation and Jacobi functions for minimal immersions in $\mathbb{S}^3$ and CMC $1$ immersions in $\mathbb{R}^3$

We give a first-order Frobenius description of conjugate minimal immersions in the round three-sphere and use it to define the conjugate of an arbitrary Jacobi function. The construction is intrinsic at the linearized level and does not require the Jacobi function to be integrable through a normal deformation by minimal surfaces. We then describe how the boundary conditions are exchanged under conjugation. Along a great-circle boundary arc, a Dirichlet Jacobi function has, after a suitable ambient gauge, a conjugate Jacobi function with vanishing Neumann data. Conversely, along a reflection curve, vanishing Neumann data yield vanishing Dirichlet data for the conjugate modulo a Jacobi--Killing function. We also construct conjugate Jacobi functions for conjugate CMC~$1$ immersions in $\mathbb{R}^3$ and exhibit cylindrical counterexamples to both pure boundary exchanges, even modulo ambient Killing fields.

math.DG↗
arXiv2026

The spherical cap conjecture for immersed disks

We prove that every smooth immersed disk in Euclidean three-space with nonzero constant mean curvature, regular up to the boundary and mapping its boundary diffeomorphically onto a circle, is an embedded spherical cap.

math.DG↗
arXiv2026

Genus two embedded minimal surfaces in $\mathbb{S}^3$ with dihedral symmetry

We prove that the Lawson surface $ξ_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in $\mathbb{S}^3$ whose isometry group contains the dihedral group $D_4$ generated by two reflections across orthogonal totally geodesic two-spheres and a half-turn about a great circle. This weakens the full-symmetry hypotheses in previous characterizations of Lawson surfaces and leads to a substantially different geometric problem.

math.DG↗
arXiv2026

Elliptic special Weingarten surfaces of minimal type in $\mathbb{R}^3$ of finite total curvature

We study complete connected embedded elliptic special Weingarten surfaces of minimal type ($f$-\emph{surfaces}, for short) in $\mathbb{R}^3$ with finite total curvature, under the standing assumption that $f$ is a non-negative and uniformly elliptic function. First, using the recent asymptotic theory of Barbieri, Gálvez, Lian, and Zhang for embedded ends, we derive a Jorge--Meeks type formula for this class of surfaces. Next, we adapt the Alexandrov reflection method to the non-cylindrically bounded setting and prove a Schoen-type theorem: a complete connected embedded $f$-surface of finite total curvature with two embedded ends must be rotationally symmetric. In particular, it is one of the special catenoids constructed by Sa Earp and Toubiana. This gives a positive answer to a question raised by Sa Earp in 1993. As a consequence, we show that planes and special catenoids are the only complete connected embedded $f$-surfaces of finite total curvature whose absolute total curvature is less than $8π$.

math.DG↗
arXiv2026

Closed $4$--Manifolds Foliated by Hyperplanes

Let $M^4$ be a closed, orientable $4$--manifold carrying a transversely oriented $C^2$ codimension--one foliation whose leaves are diffeomorphic to $\mathbb{R}^3$. We prove that $M^4$ is homeomorphic to the $4$--torus $\mathbb{T}^4$. We also show that, whenever the original smooth structure on $M$ admits a smooth defining $1$--form, the conclusion sharpens to a diffeomorphism $M\cong\mathbb{T}^4$.

math.GT↗
arXiv2026

Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow, and as a consequence we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become star-shaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend these results to the non-compact setting in two cases. First, assuming the asymptotic boundary of the solution consists of a single point, we show that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere. Second, assuming the asymptotic boundary consists of two points, we prove that the flow eventually becomes a global graph over a hyperbolic cylinder with uniform gradient bounds; this is achieved through an explicit cylindrical barrier construction analogous to the horospherical one.

math.DG↗
arXiv2026

Comparison methods for semilinear elliptic problems on Riemannian manifolds with a Ricci lower bound

In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $Δu+f(u)=0$ on domains $Ω\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci lower bound $\operatorname{Ric}_g\ge (n-1)k\,g$. Assuming admissibility and structural conditions on $f$, we prove a sharp pointwise gradient comparison, with a rigid characterization of the equality case. As applications, we derive an explicit isoperimetric-type inequality and a quantitative hot-spot localization estimate under natural convexity assumptions. In the second part, on $\mathbb S^n$ we show that isoparametric foliations produce non-rotational $f$-extremal domains, and that these examples descend to smooth quotients under free isometric actions preserving the foliation.

math.DG↗
arXiv2026

Extremal domains in $\mathbb{S}^2$: Geometric and Analytic methods

In this article, we study domains $Ω\subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ Δu + f(u,|\nabla u|)=0 \quad \text{in } Ω, $$ subject to the boundary conditions $u=0$ on $\partialΩ$ and $|\nabla u|$ being locally constant along $\partialΩ$. We refer to such domains as $f$--extremal domains. In the first part of the paper, we extend the moving plane method in $\mathbb{S}^2$ and show that if an $f$--extremal domain $Ω$ contains a simple curve of maximum points of $u$, then both $Ω$ and $u$ are either rotationally symmetric or antipodally symmetric. Using the Alexandrov reflection method, we establish an analogous symmetry result for properly embedded constant mean curvature (CMC) surfaces with capillary boundaries that contain a simple curve of minimum distance to the origin (a neck). In the second part, we strengthen these conclusions for specific nonlinearities, including the eigenvalue problem ($f(x)=λx$), the Serrin problem ($f(x)=λx + c$), harmonic domains ($f(x)=c$), and nonlinearities of the form $f(x)=λx + x^β$, for constants $λ\geq 2$, $c \geq 0$, and $β\in (0,1)$. In these cases, we prove that the domain $Ω$ must be rotationally symmetric. Throughout the paper, we restrict our attention to the analytic setting in order to simplify the exposition and highlight the main ideas.

math.AP↗
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↗
arXiv2024

Frankel property and Maximum Principle at Infinity for complete minimal hypersurfaces

In this paper, we study complete minimal hypersurfaces in Riemannian $n-$manifolds $\mathcal{M}^n$ for dimensions $4 \leq n \leq 7$, and we obtain some results in the spirit of known work for $n=3$. Key contributions include extending the work of Anderson and Rodríguez to higher dimensions. Specifically, we show that in four-dimensional manifolds with nonnegative sectional curvature and positive scalar curvature, two disjoint properly embedded minimal hypersurfaces bound a slab isometric to the product of one hypersurface with an interval. Our results are grounded in a maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces in complete Riemannian manifolds of bounded geometry, generalizing the work of Mazet in dimension three to higher dimensions. We also leverage the recent classification of complete two-sided stable minimal hypersurfaces by Chodosh, Li, and Stryker.

math.DG↗
arXiv2023

An overdetermined eigenvalue problem and the Critical Catenoid conjecture

We consider the eigenvalue problem $Δ^{\mathbb{S}^2} ξ+ 2 ξ=0 $ in $ Ω$ and $ξ= 0 $ along $ \partial Ω$, being $Ω$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere $\mathbb{S}^2$. Imposing that $|\nabla ξ|$ is locally constant along $\partial Ω$ and that $ξ$ has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.

math.AP↗
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↗
arXiv2017

Characterization of $f$-extremal disks

We show uniqueness for overdetermined elliptic problems defined on topological disks $Ω$ with $C^2$ boundary, i.e., positive solutions $u$ to $Δu + f(u)=0$ in $Ω\subset (M^2,g)$ so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte $ along $\partial Ω$, $\vecη$ the unit outward normal along $\partialΩ$ under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem to the realm of overdetermined elliptic problem. When $(M^2,g)$ is the standard sphere $\mathbb S^2$ and $f$ is a $C^1$ function so that $f(x)>0$ and $f(x)\ge x \, f'(x)$ for any $x\in\mathbb R_+^*$, we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki-Caffarelli-Nirenberg conjecture in $\mathbb S^2$ for this choice of $f$. More precisely, this shows that if $u$ is a positive solution to $Δu + f(u) = 0$ on a topological disk $Ω\subset \mathbb S^2$ with $C^2$ boundary so that $u = 0$ and $\frac{\partial u}{\partial \vecη} = cte $ along $\partial Ω$, then $Ω$ must be a geodesic disk and $u$ is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains, also called {\it Serrin Problem}) in $\mathbb S ^2$.

math.AP↗
arXiv2016

The Abresch-Rosenberg Shape Operator and applications

There exists a holomorphic quadratic differential defined on any $H-$ surface immersed in the homogeneous space $\mathbb{E}(κ,τ)$ given by U. Abresch and H. Rosenberg, called the Abresch-Rosenberg differential. However, there were no Codazzi pair on such $H-$surface associated to the Abresch-Rosenberg differential when $τ\neq 0$. The goal of this paper is to find a geometric Codazzi pair defined on any $H-$surface in $\mathbb{E}(κ,τ)$, when $τ\neq 0$, whose $(2,0)-$part is the Abresch-Rosenberg differential. In particular, this allows us to compute a Simons' type formula for $H-$surfaces in $\mathbb{E}(κ,τ)$. We apply such Simons' type formula, first, to study the behavior of complete $H-$surfaces $Σ$ of finite Abresch-Rosenberg total curvature immersed in $\mathbb{E}(κ,τ)$. Second, we estimate the first eigenvalue of any Schrödinger operator $L= Δ+ V$, $V$ continuous, defined on such surfaces. Finally, together with the Omori-Yau's Maximum Principle, we classify complete $H-$surfaces in $\mathbb{E}(κ,τ)$, $τ\neq 0$, satisfying a lower bound on $H$ depending on $κ$ and $τ$.

math.DG↗
arXiv2016

Area estimates and rigidity of capillary $H-$surfaces in three-manifolds with boundary

We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic three-manifolds under the assumption of the existence of a $H-$surface realizing the area bound, in particular, under the existence of a totally umbilic $H-$surface.

math.DG↗
arXiv2015

$f$-extremal domains in hyperbolic space

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain $Ω\subset \mathbb{H} ^2$ with $C^2$ boundary whose complement is connected and supports a bounded positive solution $u$ to an overdetermined problem, assuming natural conditions on the equation and the behaviour at infinity of the solution, must be either a geodesic ball or, a horodisk or, a half-space determined by a complete equidistant curve or, the complement of any of the above example. Moreover, in each case, the solution $u$ is invariant by the isometries fixing $Ω$.

math.AP↗
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↗