Constant mean curvature surfaces and related areas of submanifold theory represent a classical field of research that uses techniques from both Differential Geometry and Geometric Analysis. The aim of this $\mathcal{H}$-workshop is to gather together some distinguished geometers to discuss some cutting-edge discoveries in this field, as well as to give PhD students the opportunity to present their works. This event has been conceived as part of the project constant mean curvature surfaces in homogeneous 3-manifolds, supported by MICIN/AEI grant PID2022-142559NB-I00.
Registration
Everyone is invited to join the $\mathcal{H}$-workshop 2026. If you would like to attend the conference, please email us to or , so that we can estimate the number of participants.
List of confirmed speakers
abstract
Ildefonso Castro-Infantes
Universidad de Murcia
Helicoidal surfaces in the Lorentz-Minkovski 3-space with prescribed mean curvature.
Abstract:
n this talk, we investigate the geometry of helicoidal and, specifically, rotational surfaces within the three-dimensional Lorentz-Minkowski space $\mathbb{L}^3$. Using the linear momentum of the generatrix curves as key tool, we establish several existence and characterization results for such surfaces under a prescribed mean curvature function. This approach allows for a systematic treatment of the underlying differential equations, leading to a broader classification of these geometric configurations. The results presented are part of an ongoing research project in collaboration with Ildefonso Castro and Paula Carretero.
abstract
Jesús Castro-Infantes
Universidad Politécnica de Madrid
Genus one CMC unduloids with bi-dihedral symmetry
Abstract:
In this talk, we study the moduli space of genus-one unduloids with bi-dihedral symmetry in $\mathbb R^3$, that is, $H$-surfaces of genus one with embedded ends asymptotic to unduloids. To this end, we analyze the conjugate sister surface of the fundamental piece, which is a minimal surface in $\mathbb{S}^3$ via the Lawson correspondence. In particular, we prove that for each $t \in (0, k\pi - 2\pi)$ there exist at least two genus-one unduloids with $k \ge 3$ ends whose core curve of the genus-one handle has length $t$. The limiting case $t = 0$ corresponds to a configuration of $k$ spheres collapsing to a single point, whereas the limit $t = k\pi - 2\pi$ corresponds to $k$ mutually tangent spheres. This is joint work with J. M. Manzano and J. S. Santiago.
abstract
Alberto Cerezo
Universidad de Castilla-La Mancha
Capillary minimal annuli in $\mathbb{B}^3$ and an overdetermined problem in $\mathbb{S}^2$
Abstract:
For every $n \geq 2$, we show the existence of a one-parameter real-analytic family $\{\mathbb{A}_n(a) : a \in (0,1)\}$ of non-rotational embedded capillary minimal annuli in the Euclidean unit ball $\mathbb{B}^3$. Each family interpolates between a capillary catenoid (as $a \to 1$) and a necklace of $n$ capillary disks in $\mathbb{B}^3$ (as $a \to 0$). This result provides an affirmative answer to a conjecture posed by Fernández, Hauswirth, and Mira. As a corollary, we show that the capillary analogue of the critical catenoid conjecture fails for contact angles arbitrarily close to $\frac{\pi}{2}$.
We further use these annuli to construct families of non-radial ring domains $\{\Omega_n(a) : a \in (0,1)\}$, $\Omega_n(a) \subset \mathbb{S}^2$, that admit solutions to the overdetermined problem $\Delta u + 2u = 0$ in $\Omega_n(a) \subset \mathbb{S}^2$ with constant Dirichlet and Neumann data. This yields a negative answer to a 2005 conjecture of Souam. From an analytic viewpoint, these families can be interpreted as global bifurcation branches emerging from the set of rotational bands in $\mathbb{S}^2$.
Andrea Del Prete
Universiad de Jaén
TBA
José María Espinar
Universidad de Granada
TBA
Jorge Hidalgo
Universidad de Granada
TBA
Elena Mäder-Baumdicker
Freie Universität Berlin
TBA
abstract
Diego A. Marín
Universidad de Granada
Geometry of $f$-extremal domains in the 2-sphere
Abstract:
Given a riemannian manifold $(M,g)$ and a Lipschitz function $f \in \mathcal{C}(\mathbb{R}^2)$, we say that a domain $\Omega \subset M$ with $\mathcal{C}^2$-boundary is an $f$-extremal domain if it supports a solution to the overdetermined elliptic problem (OEP) \begin{eqnarray} \label{OEP}\left\{\begin{array}{llll}\Delta{u} + f(u, |\nabla u|) = 0 &\mathrm{in}~\Omega,\\u > 0 &\mathrm{in}~\Omega, \\u = 0 &\mathrm{on}~\partial\Omega,\\ \langle\nabla{u},\eta\rangle = \alpha_i &\mathrm{on}~\Gamma_i \subset \partial \Omega,\end{array}\right.\end{eqnarray} where $\eta$ is the outer conormal to $\partial \Omega$ and $\alpha_i \leq 0$ is a constant on each connected component $\Gamma_i \subset \partial \Omega$.
It is a well known fact in the literature that the theory of the existence and rigidity of $f$-extremal domains is closely linked to the theory of constant mean curvature surfaces (CMC) in $M$.
In this talk, we will explore this connection for $f$-extremal domains in $\mathbb{S}^2$. In particular, using techniques from the theory of CMC-surfaces, we will prove that, under certain conditions on the function $f$ and the topology of the domain $\Omega$, an $f$-extremal domain in $\mathbb{S}^2$ must exhibit significant symmetry. If time permits, we will also show how these techniques can be applied to study capillary CMC surfaces in the unit ball, a topic that has received a great deal of attention in recent years.
This talk is based on joint work with my supervisor, José M. Espinar.
Pablo Mira
Universidad Politécnica de Cartagena
TBA
abstract
Ivan Miranda
Instituto de Matemática Pura e Aplicada
Finite index constant mean curvature hypersurfaces in low dimensions
Abstract:
We prove that every complete finite index immersed CMC hypersurface is either minimal or compact, provided that the ambient six-dimensional manifold is a Riemannian product of a closed manifold with non-negative sectional curvature and a Euclidean factor. This answers affirmatively a question posed by do Carmo, for this class of ambient spaces, and extends known lower dimensional results. As a consequence, we complete the classification of two-sided, complete weakly stable CMC hypersurfaces immersed in the space forms of positive curvature in dimension six. We also prove that a complete, finite index CMC hypersurface immersed in the hyperbolic six-space with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.
abstract
Claudia Pontuale
Università degli Studi dell'Aquila
Stability of CMC hypersurfaces and Do Carmo's problem
Abstract:
Do Carmo’s problem asks whether a complete, noncompact, stable hypersurface with constant mean curvature in Euclidean space must necessarily be minimal. After briefly recalling some known results, we present a partial answer under a natural assumption on the Ricci curvature. The result applies in arbitrary dimension. This is a joint work with Barbara Nelli.
Mariel Sáez
Pontificia Universidad Católica de Chile
TBA
José S. Santiago
Universidad de Jaén
TBA
abstract
Ben Sharp
University of Leeds
Embedded minimal surfaces in closed analytic 3-manifolds
Abstract:
TBA
abstract
Giel Stas
KU Leuven
Parallel mean curvature spheres in the product of the two-sphere and the hyperbolic plane
Abstract:
We discuss classification results for parallel mean curvature surfaces in the product of a two-sphere and a hyperbolic plane (with opposite sectional curvatures). In particular, we prove a reduction of codimension for PMC spheres. This is an extension of a result by Torralbo and Urbano concerning PMC surfaces in the product of two surfaces with equal constant curvatures.
Marcos Tassi
Universidad de Granada
TBA
abstract
Joeri Van der Veken
KU Leuven
On minimal homogeneous submanifolds of the hyperbolic space
Abstract:
A fundamental question in submanifold theory is the following: How does the intrinsic geometry of a Riemannian manifold influence the possible isometric immersions of that manifold?
. Of particular interest is the case where the ambient space has constant sectional curvature. In the present work we consider Riemannian manifolds that are (intrinsically) homogeneous, i.e., their isometry groups act transitively. It is conjectured that a minimal isometric immersion of such a homogeneous Riemannian manifold into a Euclidean or hyperbolic space must be totally geodesic. For hypersurfaces, this follows from classification results by Nagano and Takahashi, but there are only few results in higher codimension. In this joint work with Felippe Guimarães (Federal University of Rio de Janeiro), we prove the conjecture for minimal isometric immersions of homogeneous Riemannian manifolds of dimension at least 5 and codimension 2 in hyperbolic spaces.
abstract
Tjaša Vrhovnik
Universidad de Granada
On proper minimal surfaces
Abstract:
TBA
Zhenxiao Xie
Beihang University (Beijing)
TBA
Tentative schedule
|
Mon 15 |
Tue 16 |
Wed 17 |
Thu 18 |
| 9:05 |
opening |
Short 3 |
Short 6 |
Short 9 |
| 9:30 |
Plenary 1 |
Plenary 4 |
Plenary 7 |
Plenary 10 |
| 10:20 |
Short 1 |
Short 4 |
Short 7 |
Short 10 |
| 10:45 |
coffee |
coffee |
coffee |
coffee |
| 11:20 |
Plenary 2 |
Plenary 5 |
Plenary 8 |
Plenary 11 |
| 12:10 |
Short 2 |
Short 5 |
Short 8 |
Short 11 |
| 12:35 |
Plenary 3 |
Plenary 6 |
Plenary 9 |
Plenary 12 |
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| 20:30 |
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dinner |
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How to get to IMAG
The Maths Institute is located next to the Faculty of Social Sciences of the University of Granada. To get there, you have to access via the Faculty of Social Sciences or via the Documentation Center, both located in the street Rector López Argüeta.
