Geometric Variational Problems

Workshops

Workshop on Geometric Variational Problems in Sub-Riemannian Geometry

Granada, 10-12 September 2025, Conference Room (Mathematics building, Faculty of Sciences)

Participants

  • Cheng, Jih-Hsin
  • Citti, Giovanna
  • Franceschi, Valentina
  • Giovannardi, Gianmarco
  • Monti, Roberto
  • Pinamonti, Andrea
  • Pozuelo, Julián
  • Rigot, Séverine
  • Rizzi, Luca
  • Serra-Cassano, Francesco
  • Verzellesi, Simone
  • Vittone, Davide
  • Yu, Yonghao

Scientific and Organizing Committee

Ana Hurtado, Manuel Ritoré, and César Rosales

Schedule

WednesdayThursdayFriday
9:30-10:30
10:30-11:00
11:00-12:00
12:00-13:00
13:00-15:00
15:00-16:00
16:00-17:00

Abstracts

Jih-Hsin Cheng
CR invariant surfaces and hyperbolic equations
Abstract
I would like to introduce two CR invariant surface energies $E_1$ and $E_2$ discovered in mid-nineties and express them in terms of quantities in pseudohermitian geometry (a special kind of subriemannian geometry). The $E_2$-energy appears to be the log term coefficient in the expansion of the volume renormalization associated to the singular Yamabe problem. We study the $E_1$-minimizers while (nonnegative) $E_1$ is an analogue of the Willmore energy in conformal geometry. Surprisingly the equation $E_1=0$ is hyperbolic. We solve an initial-value problem via the theory of hyperbolic equations and classify all the solutions of rotational invariance in the Heisenberg group. This presentation includes joint works with Paul Yang, Yongbing Zhang and Hung-Lin Chiu respectively.

Giovanna Citti
A representation formula on submanifolds in the Heisenberg group
Abstract
The Laplace operator and its fundamental solution on planes in the first Heisenberg group $\mathbb{H}^1$ are studied. Both characteristic and non characteristic planes, with the distance induced by the immersion, are considered. On the plane a suitable Laplacian-type operator is introduced, whose fundamental solution is a power of the distance, and a representation formula and a Poincaré inequality for functions belonging to the natural Sobolev spaces are proven. This is joint work with Baldi, Cupini, and Galeotti.

Valentina Franceschi
Mean convex mean curvature flow in the Heisenberg group 
Abstract: The mean curvature flow (MCF) describes the evolution of a hypersurface in time, where the velocity at each point is given by its mean curvature vector (i.e., the unit normal vector multiplied by the mean curvature). When initiated with a sphere in the Euclidean space, the MCF will shrink it homothetically to a point in finite time. In this talk, we introduce an adaptation of the mean convex MCF within the Heisenberg group setting. Our initial objective was to explore potential connections between this flow and the Heisenberg isoperimetric problem.
We will discuss the existence and uniqueness of solutions and prove that the Pansu sphere does not evolve homothetically under the MCF. This work is based on joint research with Gaia Bombardieri and Mattia Fogagnolo.

Gianmarco Giovannardi
The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group
Abstract: We will deal with the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body, for t-graphs on a bounded domain $\Omega$ in the Heisenberg group $\mathbb{H}^n$ . When the prescribed datum is constant and strictly smaller than the Finsler mean curvature of the boundary of $\Omega$, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme. This is joint work with A. Pinamonti, J. Pozuelo and S. Verzellesi.

Roberto Monti
Plateau’s problem for intrinsic graphs in the Heisenberg group
Abstract: Using a geometric construction, we solve Plateau’s Problem in the Heisenberg group for intrinsic graphs defined on a convex domain $D$, under a smallness condition either on the boundary $\partial D$ or on the boundary data. The proof relies on a calibration argument by Adesi Barone, Serra, Cassano, and Vittone. The techniques can also be used to establish a regularity theorem for $H$-perimeter minimizers, which slightly improves upon recent results by Giovannardi and Ritoré. This is a joint project with Giacomo Vianello from the University of Padova.

Andrea Pinamonti
Curvature measures and the sub-Riemannian Gauss–Bonnet Theorem
Abstract
Motivated by previous results in the literature, we adopt a measure-theoretic perspective to prove a sub-Riemannian Gauss–Bonnet theorem for surfaces in 3D contact manifolds via the Riemannian approximation scheme. In particular, our results recover those obtained in earlier works, providing a reinterpretation of the zero-order term in the limit as a singular measure supported on isolated characteristic points. Furthermore, we allow for mildly degenerate characteristic points, thereby generalizing the notion introduced by Tommaso Rossi. The talk is based on a joint paper with D. Barilari and E. Bellini.

Julián Pozuelo
Title: TBA
Abstract:

Séverine Rigot
Monotone sets in Carnot groups
Abstract: Monotone sets have been introduced about fifteen years ago by J. Cheeger and B. Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into $L_1$ to the classification of its monotone subsets. In this talk, I will explain further motivations for studying monotone sets in the wider setting of Carnot groups. They stem from geometric measure theory issues, as well as their relationship with minimal hypersurfaces and subharmonic functions. I shall also explain measure-theoretic and topological properties of monotone sets and give some hints towards their classification in this wider setting. Parts of the talk are based on joint works with E. Le Donne and D. Morbidelli.

Luca Rizzi
Title: Measure contraction properties for sub-Riemannian structures beyond step 2
Abstract
The measure contraction property (MCP in short, introduced by Ohta and Sturm) is the only “classical” synthetic Ricci bound that can be satisfied by sub-Riemannian structures (that are not Riemannian). Thanks to several contributions in the past 15 years, the qualitative picture about the validity of this property is well-understood for self-similar spaces (in particular, Carnot groups) without non-trivial Goh abnormals. We present recent results, in collaboration with Samuel Borza (Wien), showing that MCP can fail in the presence of non-trivial Goh abnormals. In particular we prove that any Carnot group admitting a suitable quotient to the flat Martinet structure does not satisfy any MCP. An explicit algebraic characterization is provided. In particular, the Engel group, the Cartan group (and several other explicit structures of step greater than 3) do not satisfy any MCP. To prove these result, we introduce a weaker version of the essential non-branching assumption that is implied by the validity of the minimizing Sard conjecture, that has independent interest.

Francesco Serra-Cassano
The Bernstein problem for area-minimizing intrinsic graphs in the sub-Riemannian Heisenberg group
Abstract
We will deal with the so-called Bernstein problem for area-minimizing intrinsic graphs in the first Heisenberg group $\mathbb{H}^1\equiv (\mathbb{R}^3, ·)$, understood as a Carnot group and equipped with the sub-Riemannian metric structure. More precisely, the problem reads as follows: if the intrinsic graph $\Gamma_f\subset\mathbb{H}^1$ of a function $f : \mathbb{R}^2\to\mathbb{R}$, that is \[ \Gamma_f=\{(0,y,t)\cdot (f(y,t),0,0):(y,t)\in\mathbb{R}^2\} \] is (locally) area minimizing in $\mathbb{H}^1$ then, must $\Gamma_f$ be a plane, in the geometry of $\mathbb{H}^1$? We will discuss about positive and negative answers to this problem, taking the regularity of $f$ into account. In particular we will present a new positive answer to this problem when $f \in W^{1,q}_{loc} (\mathbb{R}^2)$ with $q > 4$ and $\exp(k|\partial_tf |) \in L^1_{loc}(\mathbb{R}^2)$ for positive constant $k > 0$, in collaboration with S. Nicolussi Golo and M. Vedovato.

Simone Verzellesi
Rigidity results for complete, stable hypersurfaces in sub-Riemannian Heisenberg groups
Abstract
In this talk we examine minimal hypersurfaces in sub-Riemannian Heisenberg groups, in connection with the sub-RIemannian Bernstein problem. After a survey of the known results, we propose an approach based on the celebrated work of Schoen, Simon and Yau. Namely, we provide sub-Riemannian extensions of Simons’ formula and Kato’s inequality, and we apply them to obtain integral curvature estimates for stable hypersurfaces. These results lead to structural conditions that imply a Bernstein-type rigidity theorem for smooth, non-characteristic hypersurfaces in the second Heisenberg group. This talk is based on joint works with Gianmarco Giovannardi and Andrea Pinamonti.

Davide Vittone
Differentiability results for intrinsic graphs in Heisenberg groups
Abstract: Intrinsic graphs in Carnot groups were introduced by B. Franchi, R. Serapioni and F. Serra Cassano as a tool for the study of submanifolds with (some) intrinsic regularity. After a gentle introduction to the topic, I will discuss some differentiability results (existence of a tangent plane) for such graphs: a Rademacher-type theorem for intrinsic Lipschitz graphs in Heisenberg groups together with a counterexample in general Carnot groups, and then a recent Stepanov-type differentiability result in Heisenberg groups. The talk is based on joint works with M. Di Marco, A. Julia, S. Nicolussi Golo, A. Pinamonti and K. Zambanini.

Yonghao Yu
Title: Blow-ups of minimal surfaces in the Heisenberg group
Abstract
Minimal surfaces in the Heisenberg group $\mathbb{H}^n$ are central objects in sub‑Riemannian geometry. Motivated by De Giorgi’s regularity theory for Euclidean perimeter minimizers and Monti’s extension of this theory to $\mathbb{H}^n$, we revisit the blow‑up analysis of $\mathbb{H}$-perimeter minimizing sets. Monti showed that after rescaling a perimeter minimizing set by the square root of its excess, the Lipschitz approximations of the blow‑ups converge (in $L^2$) to a limit function for dimensions $n \geqslant 2$. His analysis claimed that this limit function is independent of the first coordinate and satisfies an equation involving the Kohn‑Laplacian; however, both claims are incorrect due to a small calculation error. We show that the horizontal Laplacian of the limit function $\varphi$ does depend on the first coordinate and satisfies a new partial differential equation. Specifically, if the original set is $\mathbb{H}$-perimeter minimizing, then $\varphi$ satisfies $\partial_{y_1}\,\Delta_0 \varphi = 0$ weakly in a neighbourhood, where $\Delta_0 = \partial_{y_1}^2 + \sum_{i=2}^n (X_i^2 + Y_i^2)$ is a reduced sub‑Laplacian. When the set is strongly $\mathbb{H}$-perimeter minimizing, $\varphi$ is harmonic for $\Delta_0$. These corrections clarify the structure of blow‑ups in $\mathbb{H}^n$ and suggest a new PDE whose regularity theory could lead to further regularity of minimal surfaces.

Short talks

Gaia Bombardieri
Self-Similar Solutions to the Mean Curvature Flow in the Heisenberg Group
Abstract: We study self-similar solutions to the horizontal mean curvature flow (HMCF) in the Heisenberg group under a mean convexity assumption, emphasizing key differences from the Euclidean case. The presence of characteristic points strongly affects the existence of self-similar solutions. As a consequence, we show that Pansu’s sphere does not evolve self-similarly under HMCF. We also derive a Heisenberg counterpart of Huisken’s soliton equation, whose solutions are indeed self-similar for the HMCF. This talk is based on joint works with Luca Capogna, Mattia Fogagnolo, and Valentina Franceschi.

Mattia Galeotti
Benamou-Brenier and Kantorovich are equivalent on sub-Riemannian manifolds with no abnormal geodesics
Abstract
We consider a sub-Riemannian manifold with no abnormal geodesics and we prove that the Benamou-Brenier formulation of Optimal Transport between two compactly supported measures, is equivalent to the Kantorovich formulation. In particular, the optimal values are the same and are reached in both cases. For this results we use relaxation techniques and introduce weaker version of the Benamou-Brenier formulation, adapted for Young measures. This is joint work with Giovanna Citti and Andrea Pinamonti.

Giacomo Vianello
The behaviour of a free-boundary (almost) area-minimizing surface near vertices
Abstract: It is well known that a free-boundary area-minimizing surface inside a smooth container satisfies Young’s Law, namely, it meets the boundary of its container orthogonally. Much less is known when the container is non-smooth. In this talk, I will describe some recent results concerning the behavior of a free-boundary area-minimizing surface near an isolated singularity (vertex) of its container. I will show that in dimension 3, the surface must avoid the singularity (vertex-skipping), while in higher dimensions the situation is more intricate, and a relation between the «aperture» of the vertex and the avoidance of the singularity appears to emerge. This is joint work with Gian Paolo Leonardi.

Practical information

Venue

The workshop will be held in the Faculty of Sciences (Fuentenueva Campus) of the University of Granada. A map of the Faculty can be found here. All the talks will take place at the Conference Room of the Mathematics building, which is the red one above and to the right on the map.

Traveling to Granada

Granada has good air and ground connections, providing easy access to the city by airplane, bus, train, car and taxi. Detailed information is available here.

Accommodation

The workshop speakers will be hosted in Hotel Reina Cristina. For other participants, we suggest also these options:

Granada is a tourist city. We recommend you to make your reservation as soon as possible. 

Restaurants

These are some options near the Faculty:

Other options are the following: