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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.

Maciej Czarnecki (Uniwersytet Łódzki) Bisectors and foliations in complex hyperbolic space

Flujo por Curvatura Media Inversa con término forzado

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.

José Gabriel Torres (Pontificia Universidad Católica de Chile ) Flujo por Curvatura Media Inversa con término forzado

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.

Eddygledson Souza Gama (Universidade Federal do Ceará) Dominios de grafos completos en \(M \times \mathbb{R}\)

Constant mean curvature surfaces in \(\mathbb{E}(\kappa,\tau)\)-spaces

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.

José M. Manzano (ICMAT) Constant mean curvature surfaces in \(\mathbb{E}(\kappa,\tau)\)-spaces

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.

Marian Ioan Munteanu (Universitatea Alexandru Ioan Cuza) The Hopf fibration 4

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.

Marian Ioan Munteanu (Universitatea Alexandru Ioan Cuza) The Hopf fibration 3

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.

Marian Ioan Munteanu (Universitatea Alexandru Ioan Cuza) The Hopf fibration 2

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.

Marian Ioan Munteanu (Universitatea Alexandru Ioan Cuza) The Hopf fibration 1

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.

Giuseppe Tinaglia (King's College London) On the existence of translators in slabs

Caracterización de superficies isoparamétricas en curvatura constante vía superficies mínimas

Instituto de Matemáticas, Universidad Nacional Autónoma de México

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.

Gabriel Ruiz-Hernández (Instituto de Matemáticas, Universidad Nacional Autónoma de México) Caracterización de superficies isoparamétricas en curvatura constante vía superficies mínimas

Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifolds.

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.

Marco A.M. Guaraco (University of Chicago) Geometric aspects of semilinear elliptic PDEs and minimal hypersurfaces on closed manifolds.

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.

Gian Paolo Leonardi (Università di Modena) An introduction to the Cheeger problem III

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.

Gian Paolo Leonardi (Università di Modena) An introductionto the Cheeger Problem II

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.

Gian Paolo Leonardi (Università di Modena) An introduction to the Cheeger problem I

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.

Lucas Ambrozio (University of Warwick, U.K.) Classification of static manifolds

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.

Ulrich Pinkall (T.U. Berlin) Schrödinger's Smoke

A properly embedded holomorphic disc in the ball with finite area and dense boundary curve

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.

Franc Forstneric (Univerza v Ljubljani) A properly embedded holomorphic disc in the ball with finite area and dense boundary curve

On the geometry of the set of compact subsets of riemannian spaceforms

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.

Didier A. Solís Gamboa (Universidad Autónoma de Yucatán, México DF) On the geometry of the set of compact subsets of riemannian spaceforms

Registration. For organizational purposes registration is appreciated.Please register by June 15st, 2018.

Funding available. There is limited funding available for young participants (intended for Ph.D.-students and young Postdocs), which can cover the hotel costs. If you are interested please send a short application including CV to one of the organizers. Women are especially encouraged to apply. Please apply by June 1st, 2018.

The UConn Summer School in Minimal Surfaces, Flows, and Relativity is a focused one-week program for graduate students and recent PhDs in geometric analysis, from 16th to 20th, July 2018.

Mini-courses will be given by

Otis Chodosh (Princeton University)

Ailana Fraser (University of British Columbia)

Yng-Ing Lee (National Taiwan University)

Richard Schoen (UC Irvine)

Lu Wang (University of Wisconsin)

The summer school is aimed at graduate students and recent PhDs working in the field of geometric analysis. In addition to standard introductory coursework, graduate students interested in this program would have ideally studied differential geometry and partial differential equations, and including some depth of experience in at least one of the topic areas of the school. We particularly encourage female students and students from under-represented minority groups to apply. Participants must apply at MathPrograms.Org in order to attend the summer school. There is limited travel funding and lodging available for graduate students and recent PhDs. For more information, visit the Application and Funding page.

We are glad to announce that registration and applications for fundings for the CIME summer school “Complex non-Kähler geometry” that will be held in Cetraro (Italy) from the 9th to the 13th July 2018, are now open. You can apply from Jan 1, 2018 to Apr 30, 2018.

There will be 4 mini-courses:

Slawomir Dinew (Jagiellonian University) on "Pluripotential Theory on Hermitian Manifolds";

Andrei Teleman (Aix-Marseille Université) on "Compact Complex Surfaces";

Santiago Alberto Verjovsky (UNAM) on "Intersection of quadrics in C^n, moment-angle manifolds, complex manifolds and convex polytopes";

Xiangwen Zhang (University of California) on "The Anomaly flow and Hull-Strominger system”.

There will also be 3 research talks:

Jean-Pierre Demailly (Université Grenoble Alpes)

Vincent Guedj (Université Paul Sabatier)

John Hubbard (Cornell University)

We warmly encourage graduate students and young researchers to participate.

Join us in celebrating Brian White's long career of highly original contributions to Geometric Measure Theory, Minimal Surfaces, and Mean Curvature Flow.

Dear Researchers, We will be honored to organize 16th International Geometry Symposium on July 4-7, 2018 hosted by Manisa Celal Bayar University. It will be a great pleasure for us to welcome you in Manisa, an education land with its madrasas where many civilizations settled throughout the history and princes of Ottoman grew. We expect your participation and contribution to symposium including studies in the domain of Geometry and Geometry Education in the light of scientific improvements. We are looking forward to seeing you in Manisa… Best Regards, Prof. Dr. Mustafa KAZAZ Head of Organizing Committee

We are glad to announce that registration and applications for fundings for the CIME summer school “Geometric Analysis” that will be held in Cetraro (Italy) from the 18th to the 22nd June 2018, are now open. You can apply from Jan 1, 2018 to Apr 30, 2018.

There will be 4 lectures:

Ailana Fraser (Univ British Columbia, Canaday) on "Minimal surfaces and extremal eigenvalue problems on surfaces";

André Neves (Univ. Chicago, USA) on "Min-max Theory. Volume spectrum and Minimal surfaces";

Peter Topping (Univ. Warwick, UK) on "Ricci flow and Ricci limit space";

Paul Yang (Princeton University, USA) on "CR Geometry in three dimensions”.

Lorentzian Geometry is a branch of Differential Geometry with roots in General Relativity and ramifications in many mathematical areas: Geometric Analysis, Functional Analysis, Partial Differential Equations, Lie groups and algebras... Several groups researching on this topic throughout the world have maintained a regular series on international meetings celebrated every two years in Spain, Italy, Brazil and Germany. The edition to be held at the Banach Center will cover topics on pure and applied Lorentzian Geometry such as geometry of spacetimes, solitons, black holes, Einstein equations, geodesics or submanifolds. Poland is a perfect place for our conference, due to the existence of a very strong General Relativity groups, with many people working in Mathematical Relativity. Some of them (Bizon, Jezierski, Kijowski, Nurowski) are organizers of the conference, members of the Scientific Committee, or speakers, while other Polish relativists have expressed their interest in the participation in the meeting.

Geometric analysis represents one of the currently most active and exciting areas of mathematics. It lies at the intersection of differential geometry, analysis, partial differential equations and mathematical physics, and is having a profound impact on all of these fields, leading to the resolution of many conjectures as well as stimulating important new avenues of research. The workshop will focus on three of the most active subareas of the subject, namely geometric flows, variational methods and mathematical relativity. It will also cover the general topic of manifolds with controlled Ricci tensor, which has not only seen spectacular progress over the past few years, but which is currently accumulating many promising links with geometric flows and a wide array of applications. This workshop will bring together the leaders in the field, and make their interaction and ideas accessible to a wide audience, particularly of UK mathematicians. The UK has seen an impressive rise in geometric analysis over the past decade, and the training of PhD students and postdocs is beginning to accelerate. The workshop will provide first-hand access for these students and postdocs to the forefront of current research. This workshop is jointly funded by ICMS and the Singularities of Geometric Partial Differential Equations EPSRC programme grant.

There are a number of spaces available at the workshop for public applicants.

The Workshop on Differential Geometry of the Institute of Mathematics of the Federal University of Alagoas has become a traditional event that takes place every year in Maceio-Alagoas, during the Brazilian summer. The aim of this workshop is to gather in Maceio national and international researchers of high scientific level in the field of differential geometry. In this 8th edition the event will be held in Hotel Ponta Verde at Praia do Francês.

This 5-day workshop is addressed to graduate students and researchers in analysis, differential geometry and theoretical physics. It will mainly focus on the geometry of Lorentzian manifolds and on hyperbolic equations (Einstein, Yang-Mills,...) on these manifolds.

The 20th School of Differential Geometry will be held in João Pesssoa, Paraíba, from 27th February to 3rd March, 2018 and it is organized by the Department of Mathematics of the Federal University of Paraíba. The venue is the Cabo Branco Station of Sciences, Culture and Arts, a convention facility quite well located in the touristic urban seashore. The school is the major biennial Brazilian event in Differential Geometry with a massive participation of both Brazilian and international researchers and students. One of its main goals is to foster scientific interchange between national and international geometers. The program includes plenary talks by distinguished invited speakers as well as contributed talks selected by the scientific committee among the proposals submitted by applicants. Basic and advanced minicourses will be offered as a traditional complementary activity of scientific formation.

Professor Xavier Cabré (ICREA and Universitat Politécnica de Catalunya, Barcelona, Spain) will give a 6-hour course on "Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions".

The Lectures will take place in Departamento de Matemáticas of the Universidad Autónoma de Madrid, with the following schedule:

The wealth of mathematics to which Riemann surfaces and algebraic curves are central, as a tool is now stunning, in differential geometry, topology, algebraic geometry, singularities, mathematical physics, dynamics, hyperbolic geometry and other subjects constantly developing new techniques to work with curves, and applying them in ever changing and evolving directions. The Iberoamerican Congresses on Geometry (ICG’s) are unique in bringing people from these diverse mathematical communities together, and fostering an exchange of ideas among mathematicians in different fields, united by Riemann surfaces and related constructions.

The 7th congress will showcase the recent advances in a broad range of geometric subjects. The program consists of nine plenary talks, seven special sessions and a poster session. Plenary talks, about current issues and of historical interest, are by experts in such core areas traditionally represented at the ICG’s as Teichmüller theory, Riemann surfaces, abelian varieties, dynamics, and foliations, but also in more differential-geometric pursuits of minimal surfaces and study of min-max surfaces, and topics in probability, tropical geometry, and mathematical physics. Topics of the special sessions are algebraic surfaces, abelian varieties, hyperbolic geometry and Teichmüller theory, algebraic and complex geometry, topology of singularities, geometry and physics, and holomorphic and algebraic foliations. Only two special sessions will run in parallel, and each session consists of talks of 40 minutes.

Instituto de Matemáticas de la Universidad de Granada (IEMath-GR)

Conferenciantes confirmados: Alessandro Carlotto (ETH Zurich) Lynn Heller (Leibniz University Hannover) David Hoffman (Stanford University) Miyuki Koiso (Kyushu University) Ernst Kuwert (University of Freiburg) Rafe Mazzeo (Standford University) William H. Meeks III (UMass at Amherst) Aurea Quintino (Universidade de Lisboa) Pascal Romon (Université Paris-Est Marne-la-Vallée) Antonio Ros (University of Granada) Tristan Riviere (ETH Zurich) Andreas Savas-Halilaj (Leibniz University Hannover) Ben Sharp (University of Warwick) Giuseppe Tinaglia (King's College London) Brian White (Stanford University)