Finsler geometries can be associated with the motion of point particles in effective field theories incorporating hypothetical violations of the foundational Lorentz symmetry of relativity. Methods to construct these Finsler spaces are presented, and some of their properties are explored. This talk is summarizing the results in my recent paper, entitled “Riemann-Finsler Geometry and Lorentz-Violating Scalar Fields” written with Alan Kostelecky,
https://doi.org/10.1016/j.physletb.2018.10.011.

The talk will survey progress over the past several years (most recent of which joint work with Costante Bellettini) on the question of regularity and compactness for a large class of uniformly area bounded, mean curvature controlled hypersurfaces (codimension 1 integral varifolds) of a Riemannian manifold. Subject to the requirements that the orientable embedded parts of the hypersurfaces have mean curvature prescribed by an ambient \(C^{1, \alpha}\) function \(g\) and have Morse index (with respect to the relevant functional determined by \(g\) ) bounded uniformly, and additionally that the hypersurfaces satisfy two necessary structural conditions, the work provides a sharp size upper bound for the singular set \(S\) which says that \(S\) has codimension at least 7. In the case that \(g\) is a constant and the hypersurface is weakly stable, curvature estimates (joint work with Costante Bellettini and Otis Chodosh) follow a posteriori. The theory to be discussed builds on, generalises and unifies the pioneering work in regularity theory of many including De Giorgi, Federer, Almgren, Simons (on locally area minimizing currents), and Schoen-Simon-Yau and Schoen-Simon (on stable hypersurfaces with small singular sets) dating back to the period 1960-1980.

We prove that the stability index of a compact constant mean curvature (CMC) surface in the Euclidean space or in the unit sphere is bounded from below by a linear function of its genus. We also will discuss some results in the case of free-boundary CMC surfaces in a mean convex body. These results are part of joint works with Darlan de Oliveira.

I will review a recently appeared joint article [Math. Z. 290 (2018), 221-250] with Jean-Baptiste Casteras and Jaime Ripoll on the asymptotic Plateau problem on Cartan-Hadamard (abbr. CH) manifolds satisfying the so-called strict convexity (abbr. SC) condition. We consider CH manifolds whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to \(-\infty\) and upper bound to \(0\) simultaneously at certain rates, or on some manifolds whose sectional curvatures go to \(-\infty\) faster than any prescribed rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with \(\mathbb{Z}_2\)- multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded \((k-1)\)- dimensional submanifold of the sphere at infinity, \(2\le k\le n-1\), as the boundary data. These generalize previous results of Anderson, Bangert, and Lang.

El problema de Calabi-Yau embebido consiste en saber si una superficie mínima completa y embebida en \(\mathbb{R}^3\) ha de ser necesariamente propia. Este problema, abierto actualmente, está resuelto en algunos casos particulares, siendo el resultado más llamativo el de Colding y Minicozzi donde se asume adicionalmente que la superficie tiene topología finita (Annals of Math 2008). Daremos una idea de cómo demostrar una versión más general de este resultado, que sólo asume que la superficie tenga género finito y una cantidad numerable de finales límite. Este es un trabajo conjunto con Bill Meeks y Antonio Ros.

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, with K-contact manifolds corresponding to symplectic manifolds. It is an interesting problem to find obstructions for a closed manifold to admit such types of structures and in particular, to construct K-contact manifolds which do not admit Sasakian structures. In the simply-connected case, the hardest dimension is 5, where Kollár has found subtle obstructions to the existence of Sasakian structures, associated to the theory of algebraic surfaces. In this talk, we develop methods to distinguish K-contact manifolds from Sasakian ones in dimension 5. In particular, we find the first example of a closed 5-manifold with first Betti number \( b_1 = 0\) which is K-contact but which carries no semi-regular Sasakian structure (Joint work with J.A. Rojo and A. Tralle).

We extend a previous sharp upper bound of the first strong stability eigenvalue due to Alías et al. [1] for the context of closed submanifold immersed with nonzero parallel mean curvature vector field in the Euclidean sphere and, through this result, we obtain a new characterization for the Clifford torus.

[1] L.J. Alías, A. Barros and A. Brasil Jr., A spectral characterization of the H(r)-torus by the first stability eigenvalue, Proc. Amer. Math. Soc. 133 (2005), 875–884.

In the complex hyperbolic space \(\mathbb {C}\mathbb{H}^n\) there are no hypersurfaces (of real dimension \(2n-1\)) which are totally geodesic. The hypersurfaces imitating this condition as well as possible are bisectors i.e. equidistants from pair of points. Every bisector is uniquely described by their poles i.e. two distinct points on the ideal boundary. A spane (rep. complex spine) of the bisector is the geodesic (resp. complex geodesic) joining poles. In my talk I shall formulate a local condition for a family of bisector to form a foliation of \(\mathbb{ C}\mathbb{H}^n\) and observe these foliations on the ideal boundary which has a structure of Heisenberg group. Moreover, we shall give examples of cospinal foliations and compare the situation with totally geodesic foliations of real hyperbolic space.

En esta charla presentaremos el flujo por curvatura media inversa con término forzado en su formato clásico (existencia, unicidad, regularidad, ejemplos y convergencia de hipersuperfices convexas) y una formulación variacional en el espíritu del trabajo de Huisken e Ilamanen para el flujo de curvatura media. Si el tiempo lo permite discutiremos una generalización del resultado de Y. Liu sobre la convergencia de hipersuperficies convexas.

This is the second and last part of a 6-hour PhD mini-course on constant mean curvature surfaces in \(\mathbb{E}(\kappa,\tau)\)-spaces.

Dada \(M\) una variedad de Riemann de dimensión \(n\) y \(\Omega\) un dominio en \(M\) con frontera regular a trozos, obtenemos condiciones necesarias para la existencia de grafos completos sobre \(\Omega\) en los siguientes casos: - Grafos mínimos - Grafos de CMC - Grafos de traslación para el flujo por la curvatura media.

In this 6-hour PhD minicourse, we will give an introduction to constant mean curvature surfaces in simply-connected Riemannian homogeneous three-manifolds with four-dimensional isometry group. These spaces are contained in a two-parameter family \(\mathbb{E}(\kappa,\tau)\), depending on two real parameters \(\kappa,\tau\in\mathbb{R}\), which includes the space forms \(\mathbb{R}^3\) and \(\mathbb{S}^3\), the product spaces \(\mathbb{H}^2\times\mathbb{R}\) and \(\mathbb{S}^2\times\mathbb{R}\), and the Lie groups \(\mathrm{Nil}_3\), \(\mathrm{SU}(2)\) and \(\mathrm{SL}_2(\mathbb{R})\) equipped with special left-invariant metrics.

Throughout the course, we will discuss some basic facts about the geometry of \(\mathbb{E}(\kappa,\tau)\)-spaces, as well as some of the most important results for constant mean curvature surfaces immersed in them. Among other topics, we will review the existence of harmonic maps and holomorphic quadratic differentials, the isometric correspondence and the conformal duality, the conjugate Plateau constructions, the Jenkins-Serrin problem, and the solution to the Bernstein problem for surfaces with critical mean curvature.

We recall some classical results on the Hopf fibration \(f:S^3 \rightarrow S^2\). We focus on the preimage of a curve gamma on \(S^2\) via the projection \(f\). It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration \(M^3(c) \rightarrow S^2(r)\), where \(M^3(c)\) is an elliptic Sasakian space form.

We recall some classical results on the Hopf fibration \(f:S^3 \rightarrow S^2\). We focus on the preimage of a curve gamma on \(S^2\) via the projection \(f\). It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration \(M^3(c) \rightarrow S^2(r)\), where \(M^3(c)\) is an elliptic Sasakian space form.

We recall some classical results on the Hopf fibration \(f:S^3 \rightarrow S^2\). We focus on the preimage of a curve gamma on \(S^2\) via the projection \(f\). It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration \(M^3(c) \rightarrow S^2(r)\), where \(M^3(c)\) is an elliptic Sasakian space form.

We recall some classical results on the Hopf fibration \(f:S^3 \rightarrow S^2\). We focus on the preimage of a curve gamma on \(S^2\) via the projection \(f\). It is known as the Hopf tube over gamma and we give some curvature properties. We point out, as application related to Physics, some developments on magnetic curves on the 3-dimensional sphere. We complete the lectures extending all these studies to the fibration \(M^3(c) \rightarrow S^2(r)\), where \(M^3(c)\) is an elliptic Sasakian space form.

In this talk I will present recent joint work with Bourni and Langford. We prove that for any \( 0< A<\pi /2\) there exists a strictly convex translating solution of mean curvature flow in \(\mathbb{R}^n\) contained in a slab of width \(\pi/cosA\) and in no smaller slab.

Las superficies isoparamétricas en un espacio de dimension tres y curvatura constante tienen curvaturas principales constantes. Veremos la siguiente caracterización: Sea $M$ una superficie tal que por cada punto pasan tres geodésicas de $M$ y para cada una de las cuales, la superficie reglada con reglas ortogonales a $M$ a lo largo de la geodésica es mínima. Entonces $M$ es isoparamétrica.

In this talk I will discuss both local and global properties of the stationary Allen-Cahn equation in closed manifolds. This equation, arising from the theory of phase transitions, has a strong connection with the theory of minimal hypersurfaces. I will summarize recent results regarding the analogy between both theories, focusing on min-max constructions. In particular, new insights into both Almgren-Pitts and Marques-Neves existence theories of minimal hypersurfaces will be discussed.

The Plateau problem arises from physics, and in particular from soap bubbles and soap films. Solving the Plateau problem means to find a surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists in giving a suitable sense to the notions of "surface", "area" and "boundary". Given \(0 < d < n\) we will consider a setting, due to Almgren, in which the considered objects are sets with locally finite d-dimensional Hausdorff measure, the functional we will try to minimize is the Hausdorff area \(H^d\), and the boundary condition is given in terms of a one-parameter family of deformations. Almgren minimizers turn out to have nice regularity properties, in particular an Almgren minimizer is a \(C^{1,\alpha}\) embedded submanifold of \(\mathbb{R}^n\) up to a negligible set, and the tangent cone to any point of such a minimizer is a minimal cone. Therefore in order to give a complete characterisation of these object we need to know how minimal cones look like. The complete list of minimal cones of \(\mathbb{R}^2\) and \(\mathbb{R}^3\) has been well known long time ago while in higher dimensions the list is far from being complete and we only know few examples. My talk will focus to a small variation of this setting which we call "sliding boundary" and to minimal cones that arise in this frame.

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

Abstract: the Cheeger problem consists in minimizing the ratio between perimeter and volume among subsets of a given set $\Omega$. The infimum of this ratio is the Cheeger constant of $\Omega$, while minimizers are called Cheeger sets. Quite surprisingly, this variational problem turns out to be closely linked to a number of other relevant problems (eigenvalue estimates, capillarity, image segmentation techniques, max-flow/min-cut duality, landslide models). After introducing some essential concepts and tools from the theory of BV functions and finite perimeter sets, we shall review some classical as well as recent results on this topic.
All lectures will be delivered at the Seminar Room in the 1st floor of the Mathematics building.
Lecture 1. March 20, 12'00–13'30. Introduction. Essentials on BV functions and finite perimeter sets.
Lecture 2: March 21, 16'00–17'30. General properties of Cheeger sets. The two-dimensional case.
Lecture 3: March 22, 12'00–13'30 Links with prescribed mean curvature equation and capillarity.

A static space-time can be described as a vacuum space-time satisfying the Einstein equations (with cosmological constant) where a global notion of time is well-defined and in which all spacial slices look equal. In more geometric terms, such important models in General Relativity can be described by a Riemannian three-manifold admitting a non-negative solution to a certain second order equation. Can we describe all such static manifolds? In this talk, I will discuss some results in that direction, specially in the case of positive scalar curvature.

Fluid simulations in Computer Graphics struggle with the problem that on the one hand in most situations of interest the real flow is dominated by very thin vortex sheets and filaments, which also are responsible for much of the fine detail of the flow. On the other hand, feasible numerical grid resolutions are unable to resolve these thin structures and result in a substantial amount of numerical viscosity. Many rather artificial remedies have been proposed for this problem.

In this talk we propose to use the equations usually reserved for quantum fluids also for the simulation of ordinary fluids. We demonstrate that this can help to overcome some of the mentioned problems. Moreover, the resulting numerical algorithm is extermely simple and efficient.

I will describe the construction of a properly embedded holomorphic disc in the unit ball of \(\mathbb{C}^2\) having the following surprising combination of properties: - on the one hand, the disc has finite area, and hence is the zero set of a bounded holomorphic function on the ball; - on the other hand, its real analytic boundary curve is everywhere dense in the sphere.

In this talk we present some interesting features of the geodesic structure of the space of compact subsets of \(\mathbb{R}^n\) and \(\mathbb{H}^n\) endowed with the Hausdorff metric. In particular, we show that such spaces are not spaces of curvature bounded from below. We further investigate connections between these spaces and the Hilbert cube.