We explain Catino, Mastrolia, and Roncoroni's proof that the hyperplanes are the only orientable stable complete minimal hypersurfaces of $\mathbb{R}^4$.

Contrary to $\mathbb{R}^n$ as ambient space, there exist compact minimal surfaces in the Euclidean n-sphere $\mathbb{S}^n$. Concerning each topological class, the variety of such examples is very limited, especially in the case of higher codimension. Regarding the latter, H. B. Lawson has shown in 1970 that every minimal immersion $\psi:\Sigma\to\mathbb{S}^3$ of a 2-manifold $\Sigma$ induces a minimal immersion $\widetilde{\psi}:\Sigma\to\mathbb{S}^5$, describing the associated so-called bipolar surface. Due to the example of the Lawson surfaces $\widetilde{\tau}_{m,k}$, analysed by H. Lapointe in 2008, we know that the topology of the bipolar surface can hereby crucially differ from the surface in the 3-sphere. In this context, we give a topological classification of the bipolar Lawson surfaces $\widetilde{\xi}_{m,k}$ and $\widetilde{\eta}_{m,k}$. Additionally, we provide upper and lower area bounds, and find that these surfaces are not embedded for $m\geq 2$ or $k\geq 2$.

A Killing submersion is a Riemannian submersion from a 3-manifold $\mathbb{E}$ to a surface $M$, both connected and orientable, whose fibers are the integral curves of a non-vanishing Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$. In this setting we give a suitable definition of the graph of a function $f\in C^\infty(U)$, where $U$ is an open subset of $M$. We study the existence and uniqueness of solutions for the Jenkins-Serrin problem on relatively compact domains of $M$ and we prove two general Collin-Krust type estimates for prescribed mean curvature graphs that extend classical result. Finally, we use these tools to prove that in the Heisenberg group there exists a unique minimal graph with prescribed bounded boundary values in every unbounded domain contained in a strip of the plane $\left\{z=0\right\}$. This talk is partially based on a joint work with J.M. Manzano and B. Nelli.

We interpret the property of having an infinitesimal symmetry as a variational property in certain geometric structures. This is achieved by establishing a one-to-one correspondence between a class of cone structures with an infinitesimal symmetry and geometric structures arising from certain systems of ODEs that are variational. Such cone structures include conformal pseudo-Riemannian structures and distributions of growth vectors (2,3,5) and (3,6). In this talk we will primarily focus on conformal structures. The correspondence is obtained via symmetry reduction and quasi-contactification. Subsequently, we provide examples of each class of cone structures with more specific properties, such as having a null infinitesimal symmetry, being foliated by null submanifolds, or having reduced holonomy to the appropriate contact parabolic subgroup. As an application, we show that chains in integrable CR structures of hypersurface type are metrizable. This is a joint work with Katja Sagerschnig.

In this talk we generalized the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer $m$, which essentially measures the number of branch points. We describe the isometry group of the most symmetric example $H_m$. The surfaces $H_m$ can also be seeing either as the unique solution to a Björling problem for an hypocycloid of $m+1$ cups if $m$ is even or as the conjugate minimal surface to the unique solution to a Björling problem for an hypocycloid of $2m+2$ cups if $m$ is odd.

Una superficie mínima tiene borde libre en la bola unidad $\mathbb{B}$ de $\mathbb{R}^3$ si intersecta ortogonalmente a $\partial \mathbb{B}$ a lo largo de su borde. En 1985 Nitsche construyó un anillo mínimo con borde libre en $\mathbb{B}$, tomando una cierta porción compacta de un catenoide; el llamado catenoide crítico. En dicho trabajo, Nitsche anunció sin demostración la unicidad topológica de este ejemplo, afirmando que todo anillo mínimo con borde libre inmerso en $\mathbb{B}$ debería ser el catenoide crítico. El objetivo de esta charla es probar que esta unicidad no es cierta. Para ello, construiremos una nueva familia de anillos mínimos con borde libre inmersos en $\mathbb{B}$, y explicaremos su geometría. Dichos ejemplos nunca están embebidos. También construiremos anillos mínimos, esta vez embebidos, que intersectan con ángulo constante a $\partial \mathbb{B}$ a lo largo de su borde, lo cual da una respuesta negativa a un problema planteado por Wente en 1995. Trabajo en colaboración con Isabel Fernández y Laurent Hauswirth.

A free boundary minimal surface in the three-dimensional unit ball is a properly immersed minimal surface in the unit ball that meets the unit sphere orthogonally along the boundary of the surface. The topic was initiated by Nitsche in 1985, derived from studies by Gergonne, Schwarz, Courant, and Lewy. Basic examples are the equatorial disk and the critical catenoid. The equatorial disk is the only immersed free boundary minimal disk in the ball up to congruence. The critical catenoid is claimed to be the only embedded free boundary minimal annulus in the ball up to congruence. Recently, the problem has been attempted using a relationship with the Steklov eigenvalue problem. In this talk, I will describe previous studies in this direction and explain my uniqueness results for the critical catenoid as the embedded free boundary minimal annuli in the ball under symmetry conditions on the boundaries.

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In this talk which is based on the joint work with Kentaro Saji given in [3], taking into account the Legendrian dualities in [2] which are extensions of the Legendrian dualities in [1], we first introduce new extended Legendrian dualities for the 3-dimensional pseudo-spheres of various radii in Lorentz-Minkowski 4-space. Secondly, by connecting all of these Legendrian dualities continuously, we construct Legendrian dual surfaces (lying in these 3-dimensional pseudo-spheres) of a spacelike curve in the 3-dimensional lightcone. Finally, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the 3-dimensional lightcone.
[1] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Moscow Mathematical Journal, 9 (2009), 325-357.
[2] S. Izumiya, H. Yildirim, Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space, Topology and its Applications, 159(2012), 509-518.
[3] K. Saji, H. Yildirim, Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone, Journal of Geometry and Physics, 104593, https://doi.org/10.1016/j.geomphys.2022.104593

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The space of null geodesics of a spacetime (a Lorentzian manifold with a choice of future) sometimes has the structure of a smooth manifold. When this is the case, it comes equipped with a canonical contact structure. I will introduce the theory for a countable number of metrics on the product $S^2\times S^1$. Motivated by these examples, I will comment on how Engel geometry can be used to describe the manifold of null geodesics, by considering the Cartan deprolongation of the Lorentz prolongation of the spacetime. This allows us to characterize the 3-contact manifolds which are spaces of null geodesics, and to retrieve the spacetime they come from. This is joint work with R. Rubio.

The eigenvalues of the Laplace-Beltrami operator on a closed Riemannian manifold are very natural geometric invariants. Although in many problems the Riemannian structure is kept fixed, the eigenvalues can be seen as functionals in the space of metrics. This is the suitable setting for the calculus of variations. In this vein, El Soufi and Ilias have characterised the metrics which are critical for the first eigenvalue among all metrics of fixed volume and among all metrics of fixed volume in a conformal class. In the talk, I will prove a similar characterisation for some critical metrics which are induced by embeddings into a fixed Riemannian manifold.

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All my works on minimal surfaces has been motivated or inspired by natural sciences, including material sciences, bio-membranes, fluid dynamics, etc. I will give an informal talk (since I’m not natural scientist) about how minimal surface theory could benefit from interdisciplinary interactions.

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Traizet’s node opening technique has been very powerful to construct minimal surfaces. In fact, it was first applied to glue saddle towers into minimal surfaces. But for technical reasons, the construction has much room to improve. I will talk about the ongoing project that addresses to various technical problems in the gluing construction. In particular, careful treatment of Dehn twist has revealed very subtle interactions between saddle towers.

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Triply periodic minimal surfaces (TPMSs) are minimal surfaces in flat 3-tori. I will review recent discoveries of new examples of TPMSs and outline future steps towards an eventual complete classification of TPMSs of genus 3.

Small variations of nonlinear dynamics by means of geometric modeling are investigated. Visualization of the dynamic process is reproduced using three-dimensional cellular automata. The dependences of the transition positions of the dynamic system between the chaotic and stable states are established. Theoretical aspects of geometric fractal approximation of the dynamic process, criteria for establishing the stable state of the system, geometric fractal derivatives and integration are considered.

In a recent joint work with David Kalaj (2021), we introduced a new Finsler pseudometric on any domain in the real Euclidean space $\mathbb R^n$ for $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with the Kobayashi pseudometric on complex manifolds. This "minimal pseudometric" describes the maximal rate of growth of hyperbolic conformal minimal surfaces in a given domain. On the unit ball, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric. I will discuss sufficient geometric conditions for a domain to be (complete) hyperbolic, meaning that its minimal pseudometric is a (complete) metric.

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I will give a pedagogical introduction to the incipient mathematical theory of globally hyperbolic supersymmetric configurations in four-dimensional Lorentzian supergravity. First, I will introduce the basics of supergravity in four dimensions as well as the notion of globally supersymmetric configuration as a solution to a system of first-order spinorial differential equations, that is, the supergravity spinorial equations. Then, I will introduce the theory of spinorial polyforms associated with bundles of irreducible real Clifford modules, which provides a convenient geometric framework to study the global geometric and topological properties of the solutions to the supergravity spinorial equations. Then, I will consider the evolution problem for globally hyperbolic supersymmetric configurations, focusing on the constraint equations, their moduli of solutions, and the construction of explicit evolution flows, which we reformulate as the supergravity flow equations for a coupled family of functions and global co-frames on a Cauchy hypersurface. This will lead us to explore in detail the case of (possibly Einstein) globally hyperbolic Lorentzian four-manifolds equipped with a parallel or Killing spinor, obtaining several results about the differentiable topology and geometry of such manifolds. Finally, I will mention several open problems and open directions for future research.

In this talk I will discuss sharp differential inequalities for the isoperimetric profile function in spaces with Ricci bounded from below, and with volumes of unit balls uniformly bounded from below. After that, I will highlight some of the consequences of such inequalities for the isoperimetric problem. After a short introduction about the notion of perimeter in the metric measure setting, I will pass to the motivation and statement of the sharp differential inequalities on Riemannian manifolds. Hence, I will discuss the proof, which builds on a non smooth generalized existence theorem for the isoperimetric problem (after Ritoré-Rosales, and Nardulli), and on a non smooth sharp Laplacian comparison theorem for the distance function from isoperimetric boundaries (after Mondino-Semola). At the end I will discuss how to use such differential inequalities to study the behaviour of the isoperimetric profile for small volumes. This talk is based on some results that recently appeared in a work in collaboration with E. Pasqualetto, M. Pozzetta, and D. Semola. Some of the tools and ideas exploited for the proofs come from other works in collaboration with E. Bruè, M. Fogagnolo, and S. Nardulli.

El objetivo de la charla es explorar la geometría compleja de superficies (orientables y compactas) para obtener una demostración del teorema de Gauss-Bonnet de forma intrínseca. Este tipo de abordaje es una buena oportunidad para introducir estructuras geométricas que se aplican en una multitud de áreas fuera de la Geometría Diferencial Clásica (e.g. Geometrías Compleja y Simpléctica); además, no se necesita ninguna familiaridad con la topología de las variedades (algo esencial al utilizar triangulaciones), solamente hace falta una buena noción de cálculo con varias variables.