| Worshop on Riemmanian Geometry and Differential Operators | Castelló from 21-th october to 24-th october 2025 |
|---|---|
| Seminar TI2328DS Department of Mathematics UJI | |
| New trends on Geometric Analysis |  |
| Universitat Jaume I | Funded by AICO (Generalitat Valenciana) and IMAC |
| Tuesday Oct.21 | Wednesday Oct.22 | Thursday Oct.23 | Friday Oct.24 | |
|---|---|---|---|---|
| 9.30-10.00 | Opening words | |||
| 10.00-10.50 | V. Miquel | A. Albujer | C. Rosales | M. Ritoré |
| 10.50-11.15 | Coffee Break | Coffee Break | Coffee Break | Coffee Break |
| 11.15-12.05 | P. Bessa | L. Guijarro | E. Cabezas | A. Hurtado |
| 12.05-12.55 | J.M. Manzano | L. Mari | A. Setti | S. Markvorsen |
| 12.55-15.00 | Lunch in Campus Garden | Lunch in Campus Garden | Lunch in Campus Garden | Farewell Lunch in Campus Garden |
| 15.00-15.50 | E. Pendás | L. Alías | M. A. Javaloyes | |
| Social Dinner |
Vicente Miquel
Abstract (of the tentative talk) : A Clifford torus is a tubular surface of constant radius around a geodesic in $\mathbb{S}^3$. These tori have constant mean curvature, therefore they are fixed points for volume preserving mean curvature flow (VPMCF). Using Fermi coordinates, these and other embedded tori in $\mathbb{S}^3$ can be described as graphs of functions defined from $\mathbb{S}^1\times\mathbb{S}^1$ to $\mathbb{R}$. The proximity of one of these tori to a Clifford torus can be defined as the proximity of the functions representing them. In this talk, we will try to show that, with respect to this concept of proximity, Clifford tori are unstable for VPMCF for a general family of functions, but stable when restricted to a special family (representing Hopf tori). Joint work with M.C. Domingo
G. Pacelli Bessa
Abstract: The mean exit time function defined on the $\delta$-tube $T_{\delta}(\mathbb{S}^{n-1})$ around any equator $\mathbb{S}^{n-1} \subseteq \mathbb{S}^{n}$ of the sphere $\mathbb{S}^{n}$, ($0 < \delta < \tfrac{\pi}{2}$), goes to infinity with the dimension, while it goes to zero, with the dimension too, when we consider it defined on the geodesic balls. Namely, the mean exit time function defined on the equatorial tubes presents a kind of concentration phenomenon or fat equator effect, as it has been described in the book [MS].
Moreover, the same concentration phenomenon occurs when we consider this mean exit time function defined on tubes around closed and minimal hypersurfaces of a compact Riemannian $n$-manifold $M$ with Ricci curvature bounded from below, $\mathrm{Ric}_{M} \geq (n-1)$. Namely, a Brownian particle that begins its random movement around a closed embedded minimal hypersurface of a compact $n$-manifold $M$ with $\mathrm{Ric}_{M} \geq (n-1)$ will wander arbitrarily close to the hypersurface for a time that approaches infinity as the dimension of the ambient manifold does so as well.
José Miguel Manzano Prego .
Abstract : We will employ a general Calabi-type duality to establish a map from the family of entire minimal graphs in $\mathrm{Nil}_3$ to the family of entire minimal graphs in $\mathrm{Sol}_3$, both Lie groups endowed with their standard left-invariant Riemannian metrics. This is done by considering (as intermediate objects) entire maximal graphs in the 3-dimensional anti-deSitter space, which can be in turn classified in terms of holomorphic quadratic differentials over the plane $\mathbb{C}$ or the unit disk $\mathbb{D}$ by the work of Wan-Au. This allows us to show that there are plenty of entire minimal graphs in $\mathrm{Sol}_3$, which provides some progress on the solution to the Bernstein problem in this space. This is a joint work with A. Del Prete and José S. Santiago.
Enrique Pendás Recondo
Abstract:Zermelo navigation is the problem of finding the fastest trajectory between two points for a moving object subject to a current (such as a boat or aircraft in the presence of wind). The classical version of this problem assumes that the speed profile of the object forms a strongly convex geometric shape, in which case the solution is always a smooth curve. In this talk, we will explore the case when the speed profile is non-convex, resulting in optimal curves with turning points, such as the zigzag trajectories seen in real-world scenarios like sailboat navigation or the flight patterns of albatrosses.
This talk is based on a joint work with Steen Markvorsen and Frederik Möbius Rygaard, from the Technical University of Denmark (see [1]).
[1] S. Markvorsen, E. Pendás-Recondo and F. M. Rygaard.
Time-dependent Zermelo navigation with tacking.
ArXiv e-prints, arXiv:2508.07274 [math.DG] (2025).
Alma L. Albujer
Abstract : Given $(M^n,g_M)$ a Riemannian manifold, let us consider the Lorentzian product manifold $M^n\times\mathbb{R}_1$, that is, the product manifold $M\times \mathbb{R}$ endowed with the Lorentzian metric $g=g_M-dt^2$.
In the case $n=2$, Albujer and Alías [AA] presented some Calabi-Bernstein results for maximal surfaces in $M^2\times\mathbb{R}$, $M^2$ being a complete surface with non-negative Gaussian curvature. In fact, they obtained both a parametric and a non-parametric version of this theorem. Furthermore, they showed that the result is no longer true when $M^2$ is the hyperbolic plane, see [A, AA]. Recently, Albujer, Cruz and dos Santos [ACS] have extended the Calabi-Bernstein result for maximal surfaces in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ stands for an $n$-dimensional ($n\geq 2$) Riemannian manifold of non-negative constant sectional curvature.
In this talk we will make a review of all the above results.
[A] A. L. Albujer. New examples of entire maximal graphs in $\mathbb{H}^n\times\mathbb{R}_1$, Differential Geom. Appl. 26 (2008), no. 4, 456–462.
[AA] A. L. Albujer and L. J. Alías. Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces, J. Geom. Phys. 59 (2009), no. 5, 620–631.
[ACS] A. L. Albujer, J. P. Cruz and F. R. dos Santos. A Calabi-Bernstein type result in $M^n(c)\times\mathbb{R}_1$, preprint, 2025.
Luciano Mari
Abstract: In this talk, we will study the behaviour of stable (or $\delta$-stable) complete minimal hypersurfaces in Euclidean space and, more generally, in manifolds $\overline{M}^{n+1}$ with non-negative sectional curvature. Are they all totally geodesic for small enough $n$? What is their topology? We will discuss some new tools discovered in joint works with X. Cabré, G. Catino, P. Mastrolia and A. Roncoroni. The guideline is that $\delta$-stable minimal hypersurfaces in ambient spaces with $\overline{\rm Sec} \ge 0$ are an example of manifolds with spectral Ricci lower bounds, a class of spaces for which we provide a generalization of Cheeger-Gromoll's splitting theorem and sharp pointwise gradient estimates for the Green kernel. Among the applications, we get a new, short proof of the Stable Bernstein Theorem in $\mathbb{R}^4$, and a characterization result for the 3D-catenoid in $\mathbb{R}^4$.
Luis J. Alias
Abstract: In this lecture we will introduce some new forms of maximum principles at infinity on complete, non-compact Riemannian manifolds. We will also show some of their applications to different geometric topics like Bernstein-type results for entire graphs and, more generally, for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field. Our results are part of our joint work with Antonio Caminha and F. Yure do Nascimento, from Universidade Federal do Ceará at Fortaleza (Brazil).
L. J. Alías, A. Caminha and F. Y. do Nascimento. A maximum principle at infinity with applications, J. Math. Anal. Appl. 474 (2019), 242–247.
L. J. Alías, A. Caminha and F. Y. do Nascimento. Spacelike hypersurfaces in conformally stationary spacetimes. RSME Springer Ser., 4, Springer, Cham, 2020, 161–174.
L. J. Alías, A. Caminha and F. Y. do Nascimento. A maximum principle related to volume growth and applications, Ann. Mat. Pura Appl. 200 (2021), 1637–1650.
César Rosales
Abstract: Given a norm $\Psi$ in $\mathbb{R}^{3}$ and a surface $\Sigma\subset\mathbb{R}^{3}$, the anisotropic area of $\Sigma$ is defined as $$A_\Psi(\Sigma)=\int_\Sigma\Psi(N)\,d\Sigma,$$ where $N$ is a unit normal over $\Sigma$ and $d\Sigma$ denotes the Euclidean area element. When $\Psi=\|\cdot\|_2$ (isotropic case) we recover the Euclidean area. A surface $\Sigma\subset\mathbb{R}^3$ is $\Psi$-minimal if it is critical for $A_\Psi$ under compactly supported deformations.
The anisotropic Bernstein problem seeks entire $\Psi$-minimal graphs. In the isotropic case it is well-known that the planes are the unique solutions. Indeed, this statement still holds for any norm $\Psi$ which is smooth outside the origin, as was proved by Jenkins (1961) by using complex variable methods. In this talk we will show a different and self-contained proof of this fact based on a differential geometric approach.
Esther Cabezas-Rivas
Abstract: We will provide several examples of past, present and future work on variational problems (including geometric flows like inverse mean curvature flow or Ricci flow or minimization problems with applications to image processing) under mild regularity assumptions, exploring existence and uniqueness issues, as well as if the solutions get smoother or keep the regularity of the source or initial data.
Alberto G. Setti
Abstract: We introduce a notion of $L^p$-parabolicity of Riemannian manifolds defined in terms of nonlinear $L^p$-capacities which extends the usual notion of parabolicity (corresponding to $p=1$) to the whole range $1\leq p\leq \infty$. $L^p$-parabolicity turns out to be equivalent to the $L^q$-Liouville property for positive superharmonic functions, where $p$ and $q$ are H"older conjugate exponents, and, when $p=2$ it coincides with the biparabolicity as defined by S.Faraji and A. Grigor'yan. We also provide a new capacitary characterization of the $L^1$-Liouville property. Finally we obtain an almost optimal volume growth conditions implying $L^p$-parabolicity for $1<p\leq 2$ as well as a sharp volume condition valid for all $1<p<\infty$ in the case of model manifolds.
Miguel Ángel Javaloyes
Abstract: How can a metric that depends on time be studied? In General Relativity, space and time are intertwined in such a way that they cannot be understood separately, and moreover, time is relative, depending on a choice of reference frame. But our goal in this lecture will be different. We will show that it makes sense to work with an absolute time and still consider time-dependent metrics. The role of length will be played by time, and the Riemannian metric will determine the velocity of objects in each direction. Geodesics will be defined as the fastest trajectories, which, by applying the relativistic Fermat principle, can be calculated as light-like geodesics on a Finsler spacetime, most generally, from a cone structure. Finally, we will demonstrate that these structures can be applied to the study of forest fires and that we can also define a curvature that measures how geodesics diverge using the degenerate curvature introduced by Harris in Lorentzian manifolds. This curvature helps to identify focal points, which are crucial for firefighters. Moreover, we will give an interpretation of curvature in terms of Jacobi fields and will obtain some applications to compute flag curvature in Finsler manifolds.
Manuel Ritoré
Abstract: We provide a simplified proof of existence of isoperimetric sets in Carnot groups with a sub-Finsler structure which is not necessarily symmetric. The proof is related to the recent one by Pozuelo in the class of nilpotent groups.
Ana Hurtado
Abstract: Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this talk is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we deduce criteria ensuring the $h$-parabolicity or $h$-hyperbolicity of $P$. Second, we employ functions with geometric meaning to describe submanifolds of bounded $h$-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems. Joint work with Vicente Palmer and César Rosales.
Steen Markvorsen
Abstract: Motivated by the physics of anisotropic conductive materials we consider a modified Laplacian operator $\Delta_{\mathcal{W}}$ of divergence type on a smooth, complete, non-compact Riemannian manifold $(M^{n}, g)$. The operator is determined by the metric $g$ and by a given \emph{conductivity}, modeled by a smooth tensor field $\mathcal{W}$ of type $(1,1)$, self-adjoint and positive definite with respect to $g$. The talk will be concerned with conditions on $g$ and $\mathcal{W}$ which will guarantee that $(M, g, \mathcal{W})$ is $\mathcal{W}$-hyperbolic -- the physical interpretation of which is that such conductive manifolds have finite effective electrical $\mathcal{W}$-resistance to infinity. The conditions will be illustrated via classical as well as new examples of $\mathcal{W}$-hyperbolic manifolds -- including a few, where the conductivity tensor is extracted from within the geometric structure of $(M, g)$ itself, e.g. as the metric equivalents of the Einstein tensor or the Schouten tensor of $(M, g)$. The presentation is based on joint work with Vicent Gimeno i Garcia, Ana Hurtado, and Vicente Palmer.
This is the question 🤔. The one that pops up in every single session of our Workshop. So, to put an end to the mystery once and for all, we decided to immortalize the answer right here, in the sacred land of the FAQ section. Well, brace yourself, because it’s not a magic spell from Harry Potter (though it definitely sounds like one). GRYOD is actually an acronym in Spanish:
“Geometría Riemanniana Y Operadores Diferenciales”
For those who don’t speak Spanish the English version is:
Riemannian Geometry and Differential Operators
So there you have it:
✨ Mystery solved. You may now continue learning without fear. ✨