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# Bisectors and foliations in complex hyperbolic space

## Maciej Czarnecki Uniwersytet Łódzki

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.

Seminario 1ª Planta, IEMATH

# Flujo por Curvatura Media Inversa con término forzado

## José Gabriel Torres Pontificia Universidad Católica de Chile

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.

Seminario de la primera planta, IEMath

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

## José M. Manzano ICMAT

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.

# Dominios de grafos completos en $M \times \mathbb{R}$

## Eddygledson Souza Gama Universidade Federal do Ceará

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.

Seminario 1ª Planta, IEMATH

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

## José M. Manzano ICMAT

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.

# The Hopf fibration 4

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

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.

# The Hopf fibration 3

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

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.

# The Hopf fibration 2

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

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.

# The Hopf fibration 1

## Marian Ioan Munteanu Universitatea Alexandru Ioan Cuza

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.

# On the existence of translators in slabs

## Giuseppe Tinaglia King's College London

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.

Seminario 1ª Planta, IEMATH

# Caracterización de superficies isoparamétricas en curvatura constante vía superficies mínimas

## Gabriel Ruiz-Hernández 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.

Seminario 1ª planta,IEMath-UGR

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

## Marco A.M. Guaraco University of Chicago

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.

Seminario 1ª Planta, IEMATH

# Minimal boundary cones

## Edoardo Cavalloto Université Paris-Sacaly

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.

Seminario 1ª Planta, IEMATH

# An introduction to the Cheeger problem III

## Gian Paolo Leonardi Università di Modena

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.

# An introductionto the Cheeger Problem II

## Gian Paolo Leonardi Università di Modena

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.

# An introduction to the Cheeger problem I

## Gian Paolo Leonardi Università di Modena

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.

Sala de conferencias, primera planta

# Classification of static manifolds

## Lucas Ambrozio University of Warwick, U.K.

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.

A25, Facultad de Ciencias

# Schrödinger's Smoke

## Ulrich Pinkall T.U. Berlin

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.

Sala de conferencias, IEMath

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

## Franc Forstneric Univerza v Ljubljani

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.

Seminario 1ª Planta, IEMATH

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

## Didier A. Solís Gamboa Universidad Autónoma de Yucatán, México DF

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.

Seminario 1ª Planta, IEMATH

# Geometric Analysis

## ICMS Edinburgh (Reino Unido)

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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.

# Spectral estimates on non-compact manifolds and applications to geometry

## MFO Oberwolfach (Alemania)

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Organised by Nadine Große (Freiburg), Luciano Mari (Pisa), and Ben Sharp (Warwick). Deadline for applications: March 19th.

# First BYMAT Conference Bringing Young Mathematicians Together

## Madrid (España)

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Welcome to the “1st BYMAT Conference: Bringing Young Mathematicians Together". This conference aims to:
• Strengthen the links between PhD students in mathematics across all disciplines.
• Provide an open space so researchers in the early stages of their career can present their work to peers of similar experience.
• Enhance the communication skills of young mathematicians.
• Encourage researchers of different institutions to start building a network of contacts soon into their careers.
• Showcase the broad range of career options available for a mathematics PhD graduate in and outside of academia.

# VIII Workshop on Differential Geometry

## Maceio (Brasil)

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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.

# Field equations on Lorentzian space-times

## Universität Hamburg (Alemania)

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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.

# 20th School of Differential Geometry

## João Pesssoa, Paraíba (Brasil)

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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.

# Lectures on PDEs and Geometry

## Universidad Autónoma de Madrid (España)

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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:
1. Tuesday, January 23 2018 Aula 520, 11:00-13:00
2. Wednesday, January 24 2018 Aula 520, 11:00-13:00
3. Thursday, January 25 2018 Aula 520, 11:00-13:00

# Seventh Iberoamerican Congress on Geometry January 22nd-26th, 2018. Valladolid, Spain

## Valladolid (España)

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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.

# Minimal surfaces and related topics

## Granada (España)

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http://gigda.ugr.es/minimal2018/

Download the Official Poster.

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)