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A semi-Riemannian manifold is geodesically complete (or for short, complete) if its maximal geodesics are defined for all times. For Riemannian metrics the compactness of the manifold implies completeness. In contrast, there Lorentzian metrics on the torus that are not complete. Nevertheless, completeness plays an important role for fundamental geometric questions in Lorentzian geometry such as the classification of compact Lorentzian symmetric spaces and in particular for a Lorentzian version of Bieberbach's theorem. We will study the completeness for compact manifolds that arise from the classification of Lorentzian holonomy groups, which we will briefly review in the talk. These manifolds have abelian holonomy and carry a parallel null vector field. By determining their universal cover we show that they are complete. In the talk we will explain this result and further work in progress, both being joint work with A. Schliebner (Humboldt-University Berlin).

Thomas Leistner (University of Adelaide) Geodesic completeness of compact Lorentzian manifolds

I will introduce Lyapounov functions for cone fields, a generalization of the causal structure of a Lorentzian metric, and present some results on their existence. If time permits I will define a notion of global hyperbolicity for cone fields and give a result on the existence of steep Lyapounov functions for globally hyperbolic cone fields. The material is a cooperation with Patrick Bernard (Paris).

Stefan Suhr (University of Bochum) Lyapounov functions for cone fields

On the topology of surfaces with the simple lift property

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface \(\Sigma\subset\Omega\) with the simple lift property must have genus zero, if \(\Omega\) is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that \(\Omega\) cannot contain closed minimal surfaces. In this work, I generalize this result by taking an arbitrary orientable three-manifold \(\Omega\) and proving that one is able to restrict the topology of an arbitrary surface \(\Sigma\subset\Omega\) with the simple lift property. Among other things, I prove that the only possible compact surfaces with the simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where \(\Sigma\subset\Omega\) is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.

Francesca Tripaldi () On the topology of surfaces with the simple lift property

Asymptotic Dirichlet problems for the mean curvature operator

In \(R^n\) (\(n\) at most 7) the famous Bernstein's theorem states that every entire solution to the minimal graph equation must be affine. Moreover, entire positive solutions in \(R^n\) are constant in every dimension by a result due to Bombieri, De Giorgi and Miranda. If the underlying space is changed from \(R^n\) to a negatively curved Riemannian manifold, the situation is completely different. Namely, if the sectional curvature of \(M\) satisfies suitable bounds, then \(M\) possess a wealth of solutions. One way to study the existence of entire, continuous, bounded and non-constant solutions, is to solve the asymptotic Dirichlet problem on Cartan-Hadamard manifolds. In this talk I will discuss about recent existence results for minimal graphs and f-minimal graphs. The talk is based on joint works with Jean-Baptiste Casteras and Ilkka Holopainen.

Esko Heinonen (Universidad de Granada) Asymptotic Dirichlet problems for the mean curvature operator

Obtenemos un teorema de caracterización del cilindro construido sobre una curva grim reaper como el único solitón de traslación del flujo por la curvatura media en el espacio euclídeo \(R^n\) que es asintótico a dos hiperplanos fuera de un cilindro. Esto generaliza a dimensión arbitraria un resultado previo de Martin-Perez-Savas-Smoczyk para solitones de \(R^3\).

Eddygledson Souza Gama (Universidade Federal do Ceará) Unicidad del cilindro grim reaper en \(R^n\)

Minimal graphs with micro-oscillations and global approximation theorems in PDEs

In the 1950s, Lax and Malgrange proved that a solution v of a linear elliptic equation with analytic coefficients Pv=0 in a compact set can be approximated by a global solution u of Pu=0 provided that the complement of the set is connected. This theorem is key to showing the existence of minimal graphs on the unit ball whose transverse intersection with a horizontal hyperplane has arbitrarily large (n-1)-measure. The proof hinges on the construction of minimal graphs which are almost flat but have small oscillations of prescribed geometry. In addition, I will present an overview of our recent generalization of the Lax-Malgrange result to the case of parabolic equations and discuss applications to the study of hot spots. This is a joint work with A. Enciso and D. Peralta-Salas.

María Angeles García-Ferrero (ICMAT) Minimal graphs with micro-oscillations and global approximation theorems in PDEs

University of Trento (Italy) and University of Jyvaskyla (Finland)

The Heisenberg group has a well studied sub-Riemannian structure. Geometric measure theory in the sub-Riemannian setting is still in development and several fundamental questions are still open. One reason is that sets of finite sub-Riemannian perimeter may have fractal behaviours. I will present the state of art (in my knowledge) of the study of minimal surfaces in this setting, with some recent results about contact variations of the area functional.

Sebastiano Nicolussi Golo (University of Trento (Italy) and University of Jyvaskyla (Finland)) Minimal surfaces in the Heisenberg group

In a lesser-known paper [Berl. Ber. 1921, pp. 261-264], Albert Einstein proposed the natural addition of a scalar differential equation to the field equations of General Relativity. I shall explore this suggestion from a geometrical point of view. As a consequence, I will show a family of scalar conformal invariants for generic (pseudo-)Riemannian manifolds of dimension greater than 3.

Ignacio Sánchez (Universidad de Granada) Scalar conformal differential invariants

Sesión II - Curso "La desigualdad riemanniana de Penrose y el flujo inverso de la curvatura media"

La demostración de la desigualdad de Riemann-Penrose por parte de Huisken e Ilmanen al final del pasado siglo fue un avance importante tanto para la Teoría de Agujeros Negros como para la Geometría Diferencial. Para la primera, tal desigualdad puede ser vista como el mejor test de su consistencia física, debido a la dificultad en observar evidencias directas de los agujeros negros. Para la Geometría Diferencial, la demostración puso de manifiesto cuán poderoso y versátil es el llamado flujo inverso de la curvatura media, incluyendo la formulación débil de las soluciones, que permite a las superficies del flujo ”saltar” más allá de las singularidades. De hecho, este flujo fue usado, poco después, por Bray y Neves para calcular el invariante de Yamabe del espacio proyectivo real de dimensión 3. El propósito de este curso es el de estudiar en detalle la demostración de Huisken e Ilmanen, proporcionando previamente una perspectiva de las motivaciones físicas y de las herramientas acerca de flujos geométricos que son necesarias para su comprensión.

Miguel Sánchez (Universidad de Granada) Sesión II - Curso "La desigualdad riemanniana de Penrose y el flujo inverso de la curvatura media"

Sesión 1 - Curso "La desigualdad riemanniana de Penrose y el flujo inverso de la curvatura media"

La demostración de la desigualdad de Riemann-Penrose por parte de Huisken e Ilmanen al final del pasado siglo fue un avance importante tanto para la Teoría de Agujeros Negros como para la Geometría Diferencial. Para la primera, tal desigualdad puede ser vista como el mejor test de su consistencia física, debido a la dificultad en observar evidencias directas de los agujeros negros. Para la Geometría Diferencial, la demostración puso de manifiesto cuán poderoso y versátil es el llamado flujo inverso de la curvatura media, incluyendo la formulación débil de las soluciones, que permite a las superficies del flujo ”saltar” más allá de las singularidades. De hecho, este flujo fue usado, poco después, por Bray y Neves para calcular el invariante de Yamabe del espacio proyectivo real de dimensión 3. El propósito de este curso es el de estudiar en detalle la demostración de Huisken e Ilmanen, proporcionando previamente una perspectiva de las motivaciones físicas y de las herramientas acerca de flujos geométricos que son necesarias para su comprensión.

Miguel Sánchez (Universidad de Granada) Sesión 1 - Curso "La desigualdad riemanniana de Penrose y el flujo inverso de la curvatura media"

The Dirichlet problem for the constant mean curvature equation in Sol3

In this talk we will present a version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol3. Moreover, we will construct examples of admissible domains where the results may be applied. This is a joint work with Patricia Klaser.

Ana Maria Menezes (Princeton University) The Dirichlet problem for the constant mean curvature equation in Sol3

Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S) >= Area(W), provided the intersections of S with horizontal planes have the same orientation.This is joint work with Jaigyoung Choe.

Benoît Daniel (Université Henri Poincaré - Nancy 1) On the area of minimal surfaces in a slab

Real hypersurfaces with isometric Reeb flow in Hermitian symmetric spaces of compact type II

Translating Solitons are some special solutions to the Mean Curvature Flow. The classical examples appear in the Euclidean Space. We review some basic properties, both in Euclidean and Minkoswki Space. Here, we will pay attention to those whose direction of translation is lightlike, in dimensions 2 and 3, obtaining some examples.

Erdem Kocakuşaklı (Universidad de Ankara) Translating Solitons in Minkowski Space

In this talk we will present the main results of a work in collaboration with S. Canevari and G. M. de Freitas. More precisely, we prove that complete submanifolds, on which the weak Omori-Yau maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.

Fernando Manfio (Universidadade de Sao Paulo) Submanifolds with nonpositive extrinsic curvature

Less than two years ago some physicists gave the official announcement that gravitational waves exist, but, from a geometrical point of view, they have always been “real objects” and their properties have been widely investigated. The aim of this talk is introducing generalized plane waves and discussing some of their properties such as geodesic connectedness and geodesic completeness.

Anna María Candela (Università degli Studi di Bari) Gravitational Waves: a Mathematical Point of View

En esta charla expondré algunas nociones básicas de la Relatividad Especial, sin apenas fórmulas, a través de sencillos diagramas en el espacio de Minkowski. Usando intuitivos argumentos geométricos, explicaré la paradigmática paradoja de los gemelos, y las menos conocidas paradojas de Bell y de la pértiga y el granero.

Daniel De la Fuente Benito (Universidad de Granada) Paradojas Relativistas

The concept of trapped surfaces was originally formulated by Penrose for the case of 2-dimensional spacelike surfaces in 4-dimensional spacetimes in terms of the signs or the vanishing of the so-called null expansions. This is obviously related to the causal orientation of the mean curvature vector of the surface, which provides a better and powerful characterization of the trapped surfaces and allows the generalization of this concept to codimension two spacelike submanifolds of arbitrary dimension n. In this sense, an n-dimensional spacelike submanifold \(\Sigma\) of an (n + 2)-dimensional spacetime is said to be future trapped if its mean curvature vector field \(\vec{H}\) is timelike and future-pointing everywhere on \(\Sigma\), and similarly for past trapped. If \(\vec{H}\) is lightlike (or null) and future-pointing everywhere on \(\Sigma\) then the submanifold is said to be marginally future trapped, and similarly for marginally past trapped. Finally, if \(\vec{H}\) is causal and future-pointing everywhere, the submanifold is said to be weakly future trapped, and similarly for weakly past trapped. The extreme case \(\vec{H} = 0\) on \(\Sigma\) corresponds to a minimal submanifold. In this lecture we consider codimension two compact marginally trapped submanifolds in the light cone of de Sitter space. In particular, we show that they are conformally diffeomorphic to the round sphere and, as an application of the solution of the Yamabe problem on the round sphere, we derive a classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space. This is part of our work in progress with Luis J. Alías (Univ. Murcia) and Marco Rigoli (Univ. Milano).

Verónica López Cánovas (Universidad de Murcia) Trapped Submanifolds in the De Sitter Spacetime

On neutral 4-manifolds (survey) and counterexamples to Goldberg Conjecture in indefinite metric spaces

Osaka City University Advanced Mathematical Institute

My talk will begin with a survey on the existence theorem of a neutral metric on a 4-manifold. The existence of a neutral metric on a 4-manifold is equivalent to the existence of a field 2-planes, and also to that of a pair of two kinds of almost complex structures. These facts are proved on the basis of a theorem of Hirzebruch and Hopf (1958), and of Donaldson’s classification of definite intersection forms (1983). I then consider the Goldberg Conjecture (1969) of indefinite metric version, and I report some counterexamples to the Conjecture of indefinite metric version. Also, future problems will be discussed.

Yasuo Matsushita (Osaka City University Advanced Mathematical Institute) On neutral 4-manifolds (survey) and counterexamples to Goldberg Conjecture in indefinite metric spaces

Isometric Reeb flow on Real Hypersurfaces in Hermitian Symmetric Spaces.

In this talk. first, from the histrorical view point we introduce some works on isometric Reeb flow on real hypersurfaces in complex projective space, complex hyperbolic space, complex two-plane Grassmannians, and complex quadrics. Next we want to give a recent classification of isometric Reeb flow on real hypersurfaces in generalized complex k-plane Grassamnnians.

Young Jin Suh (Kyungpook National University) Isometric Reeb flow on Real Hypersurfaces in Hermitian Symmetric Spaces.

Every meromorphic function is the Gauss map of a conformal minimal surface

We prove that every meromorphic function on an open Riemann surface \(M\) is the complex Gauss map of a conformal minimal immersion \(f:M\to \mathbb R^3\); furthermore, \(f\) may be chosen as the real part of a holomorphic null curve \(F:M\to\mathbb C^3\). Analogous results are proved for conformal minimal immersions \(M\to\mathbb R^n\) for any \(n>3\). We also show that every conformal minimal immersion \(M\to\mathbb R^n\) is isotopic to a flat one, and we identify the path connected components of the space of all conformal minimal immersions \(M\to\mathbb R^n\) for any \(n\ge 3\). (Joint work with Antonio Alarcón and Francisco J. López,

Franc Forstneric (Univerza v Ljubljani) Every meromorphic function is the Gauss map of a conformal minimal surface

Lawson correspondence for minimal and constant mean curvature surfaces

Lawson correspondence between minimal surfaces in \(\mathbb{S}^3\) and constant mean curvature (CMC) surfaces in \(\mathbb{R}^3\) is instrumental in building examples of surfaces, but also reveals a lot about the structure of underlying PDEs. In discrete geometry, simple definitions of CMC surfaces are not obvious (many have been given) and an analog to Lawson correspondence has eluded researchers for a long time. We will present here a natural and computation-friendly definition, based on Lax pairs, which showcases the analogy between the two types of surfaces, as well as proposes a construction method. We will try to outline the similarities and differences between the smooth and discrete realms. Joint work with Alexander Bobenko (TU Berlin)

Pascal Romon (Université Paris-Est Marne-la-Vallée) Lawson correspondence for minimal and constant mean curvature surfaces

On a solution for a fourth-order nonlinear partial differential equation in a compact manifold

The purpose of this article is to provide sufficient conditions for having a positive weak solution for a fourth-order non-linear partial differential equation in M, a compact Riemannian manifold, and this solution is harmonic on the boundary of M.

Mohammed El Aidi () On a solution for a fourth-order nonlinear partial differential equation in a compact manifold

Dear Colleagues, Ahmad El Soufi, Professor of Mathematics at the Laboratoire de Mathématiques et Physique Théorique (LMPT) at the University of Tours, passed away suddenly, last December. He was held in high regard, not only as a world class mathematician in geometry and an expert in spectral theory, but, also, as a person. His absence is keenly felt in our lab. With many of his colleagues we are organizing a workshop to honor his memory. It will take place in Tours, from Wednesday the 13th to Friday the 15th of September 2017. Looking forward to seeing you in Tours.

The Spanish-Portuguese Relativity Meetings are annual conferences on General Relativity and Gravitation, that date back to 1977. They are organized each year by one of the different groups doing research on Relativity and Gravitation in Portugal and Spain. The aim of this conference is to provide a forum for researchers and students to present their research results on the main areas of Relativity.

From August 21 till August 25, 2017 the Conference "Pure and Applied Differential Geometry - PADGE 2017" will take place at KU Leuven, Belgium. This event is organized by the KU Leuven Geometry Section approximately every five years.

Dear professor, Dear geometry researcher, We are pleased to invite you to the conference 'Differential Geometry' that will take place in the Banach Conference Center at Bedlewo, Poland, from June 18th till June 24th 2017. This conference is the sixth in a series organized in a Polish-European co-operation. It is sponsored by the Banach Center and the Warsaw Center of Mathematics and Computer Science. The purpose of this conference is to bring together researchers in geometry and to provide for them a forum to present their recent work to colleagues from different nationalities. This way we aim to stimulate discussion about the latest findings in differential geometry and to increase international collaboration. The conference will cover the following topics: - Riemannian and pseudo-Riemannian geometry, - classical differential geometry, - manifolds with special structures (Kaehler, Sasakian etc.), - submanifolds, - affine differential geometry, - relations between geometry and PDE's, - global problems, - classification results.

Geometric Analysis consists in applying PDE tools in order to prove deep theorems in Geometry. It is one of the areas mathematics with more spectacular results in the last half a century. The school is primarily aimed for Master and beginning Ph. D. students, and hence prerequisites will be kept to a minimum. The objective is to develop the tools and some working knowledge in Geometric Analysis, that is the interaction of manifold theory with PDE analysis. We hope to bring together students in analysis and geometry and establish a fruitful interaction between them. We have chosen as a guiding goal of the school, to develop the tools and explain a proof of one of the most classic and striking theorems in geometric analysis, namely the Atiyah-Singer Index Theorem. Besides the importance of the theorem, we feel that developing the tools, assembling a proof and explaining some applications (Hirzebruch-Riemann-Roch and Hirzebruch signature theorems) will help the students in learning useful techniques in geometry and analysis, and learning how they can be combined in order to prove deep results. The school will last for three weeks. The first two weeks will be devoted to develop basic techniques in geometry and analysis of PDE’s, which are needed for the proof of the theorem, but which are completely central by themselves. In the third one Atiyah-Singer index theorem will be proved, some applications will be derived, and in addition some topics of modern geometric analysis will be surveyed, in order to let the participants taste some topics of current research.

Un grupo de alumnos, amigos y colaboradores de Antonio Ros hemos organizado un par de días de charlas informales con motivo del 60 cumpleaños de Antonio. Durante esos dos días tendrá lugar una cena homenaje. Si estás interesado en participar en este homenaje, por favor contacta con Pieralberto Sicbaldi (pieralberto < at > ugr.es). El precio aproximado de la cena será de 35 Euros por persona.

Registration deadline for those requesting any financial/accommodation support: noon, Monday 13 March 2017.Registration closes: noon, Monday 3 April 2017.

Talks start: 9am on Monday 8 May

Talks end: 4pm Wednesday 10 May (tentative plan)

All talks in MS.04, Mathematics Institute, University of Warwick. Workshop dinner to be held Tuesday 9th May, 6.30pm, at The Cross, Kenilworth. We expect to have a buffet dinner at the Mathematics Institute on Monday 8th May.

Supported by EPSRC through a symposium grant and the Warwick-Imperial-Cambridge geometric analysis seminar (with Dafermos, Neves, Wickramasekera, funded by a Programme grant).

The JDG 2017 conference on Geometry and Topology is co-sponsored by Lehigh University and partially supported by NSF. It takes place in celebration of the 50th anniversary of the Journal of Differential Geometry.

The aim of this workshop is to bring together active young researchers on differential geometry in Lorentz-Minkowski space, specially on surfaces with constant curvature. The meeting addresses specially for graduate students, recent PhDs and other junior researchers, to meet and present their work to each other. The speakers are young researchers that will present their latest results and the lectures will be focused on familiarizing the participants with the background material leading up to specific problems.

Scientific Committee: Franz Pedit (UMass) Antonio Ros (UGR)

Organising Committee: Tim Hoffmann (TUM) Martin Kilian (UCC) Katrin Leschke (UoL) Francisco Martin (UGR) Katsuhiro Moriya (UTsukuba)

Confirmed Speakers: Benoît Daniel (IECL), Isabel Fernández (USeville), Leonor Ferrer (UGR), Laurent Hauswirth (UPE), Barbara Nelli (UnivAQ), Pablo Mira (UPCT), Franz Pedit (UMass), Joaquín Pérez (UGR), Ulrich Pinkall (TUB), Martin Schmidt (UMA)

The aim of the meeting is to gather some experts, including young ones, working actively in Geometric Analysis. We would like especially to bring together people with more analytic and more geometric backgrounds in order to fruitfully exchange ideas and methods for applications of one field to another. This interaction has been quite successful over the past decades, and we hope to create new occasions for continuing in this.

The aim of the conference is to bring together leading experts and researchers in nonlinear partial differential equations and in subjects of pure and applied mathematics, in which James Serrin inspired generations of Mathematicians and continues to influence their ideas. The conference will promote research, stimulate interactions among the participants and continue the tradition of the previous meetings on PDEs methods and their applications held in Perugia in the last decades

En nombre del Comité Organizador es un placer invitarte a participar en el congreso de la Real Sociedad Matemática Española que se celebrará en la Facultad de Educación de la Universidad de Zaragoza del 30 de enero al 3 de febrero de 2017. En este congreso Bienal RSME 2017 se dará a conocer la reciente investigación en matemáticas, así como estrecharemos lazos de colaboración entre distintos grupos de investigación de nuestro país. La asistencia al congreso permitirá disfrutar de esta ciencia a través de variadas actividades programadas en la ciudad de Zaragoza. Por todos estos motivos te animamos a participar en este evento y esperamos verte en la histórica ciudad de Zaragoza. Juan Ignacio Montijano Presidente del Comité Organizador