Abstracts

         Minicourses

• • Peter Petersen (UCLA, USA) 

  • Title: Rigidity of Alexandrov Spaces.

  • Abstracts:

  • Lecture 1: Alexandrov spaces will be defined and examples presented. We will explain some of the basic concepts such as tangent cones, spaces of directions. We will introduce modified distance functions and develop a calculus of such functions as a substitute for first and second variation formulas.

  • Lecture 2: We start by explaining the basic rigidity results for Alexandrov spaces that come from Riemannian geometry: Maximal diameter and splitting theorems of Toponogov. We continue by introducing some of the more advanced concepts such as gradient curves, quasi-geodesics. This leads to the gradient exponential map, a concept that is useful even for Riemannian manifolds, but has never been used for such spaces.

  • Lecture 3:  We motivate via Riemannian geometry the rigidity results for Alexandrov spaces with boundary that we are interested in and explain some new and also future work with K. Grove.

  • Fernando Galaz García (Karlsruher Institut für Technologie)

  • Title: Topology and Geometry of three-dimensional Alexandrov space.

  • Abstract: Alexandrov spaces are metric generalizations of riemannian manifolds with a uniform lower sectional curvature bound. In this mini-course I will discuss recent advances in our understanding of the geometry and topology of 3-dimensional Alexandrov spaces, including: the classification of positively and non-negatively curved closed Alexandrov 3-spaces, collapse phenomena, and the structure of Alexandrov 3-spaces with  isometric compact Lie group actions. I will also discuss some open problems.

  • Research Talks

• Jesús Angel Núñez Zimbrón (CCM-Morelia, UNAM, Mexico)

  • • Title: Maximal volume entropy of metric measure spaces with Ricci curvature bounded below.

• • Abstract:  The volume entropy h(M,g) of a compact, n-dimensional Riemannian manifold (M,g) is a geometric invariant that measures the rate of exponential growth of the volume of concentric balls in the universal cover of M. This invariant is related to several other invariants such as the simplicial volume and minimal volume. Ledrappier and Wang showed that h(M,g) is less than or equal to n-1 and that the equality holds if and only if (M,g) is a hyperbolic manifold. In this talk I will describe a generalization of this result to a class of metric measure spaces which have "Ricci curavture bounded below by K and dimension bounded above by N" in a synthetic sense. These spaces, the so called RCD*(K,N) spaces, are a subclass of the metric measure spaces with bounded Ricci curvature originally defined by  Lott-Sturm-Villani which are "infinitesimally Hilbertian" as defined by Gigli. Important examples of RCD*(K,N) spaces include Alexandrov spaces of (sectional) curvature bounded below. 

    • This result is part of a joint project with C. Connell, X. Dai, R. Perales, P. Suárez-Serrato and G. Wei. 

• Gabriel Ruiz (IMATE-Queretaro, UNAM)

• Title: Timelike surfaces in Minkowski space with a canonical null direction

• Abstract: Given a constant vector field Z in Minkowski space, a timelike surface is said to have a canonical null direction with respect to Z if the orthogonal projection of Z on the tangent space of the surface gives a lightlike vector field. For example in the three-dimensional Minkowski space: A surface has a canonical null direction if and only if it is minimal and flat. When the ambient has arbitrary dimension, if a surface has a canonical null direction and has parallel mean curvature vector then it is minimal. We give different ways for building these surfaces in the four-dimensional Minkowski space. On the other hand, we describe several properties in the four-dimensional Minkowski space.

• Gregor Weingart (IMATE-Cuernavaca, UNAM)

• Title: Einstein Metrics on Flag Manifolds

• Abstract: Einstein metrics are critical points of the Einstein-Hilbert functional governing the General Theory of Relativity, in other words Einstein metrics are possible vacua for leading theory to describe gravitation. In

• my talk I will consider the Einstein-Hilbert functional restricted to left invariant Riemannian metrics on compact homogenenous space, specifically the complex flag manifolds. Rather surprisingly the Einstein-Hilbert

• functional on the complex flag manifolds has quite a lot of critical points related to subgroups of the Weyl group associated to a homogeneous space.

• Miguel Angel García Ariza (Universidad de las Américas Puebla)

• Title: Scalar curvature in equilibrium thermodynamics

• Abstract: There have been recent efforts to retrieve physical information from thermodynamic systems in equilibrium using Riemannian geometry as a tool. One of the main trends in this direction is the so-called Ruppeiner geometry. This formalism endows the scalar curvature of certain Hessian metrics with relevant physical data, unattainable by any other known means. In this talk I will introduce Ruppeiner geometry from a rigorous point of view. I will also present some facts that render scalar curvature a rather unphysical object. 

• Rosemberg Toalá Enríquez (Universidad Autónoma de Chiapas)

• • Title: Stationarity of asymptotically flat non-radiating electrovacuum spacetimes.

• Abstract: I will show the main techniques to prove that a solution to the Einstein-Maxwell equations whose gravitational and electromagnetic radiation fields vanish at infinity is in fact stationary in a neighbourhood of spatial infinity. That is, if in adapted coordinates, the Weyl and Faraday tensors decay suitably fast and there is an asymptotically-to-all-orders Killing vector field, then this is indeed a Killing vector field in the region outside the bifurcate horizon of a sphere of sufficiently large radius. In particular, electrovacuum time-periodic spacetimes, which are truly dynamic, do not exist. This can be interpreted as a mild form of the statement: "Gravitational waves carry energy away from an isolated system".

• Armando Cabrera Pacheco (Universität Tübingen)

• Title: On asymptotically flat extensions of Riemannian manifolds in mathematical relativity

• Abstract: In the context of mathematical relativity, asymptotically flat Riemannian manifolds with non-negative scalar curvature represent time-symmetric initial data sets (satisfying the dominant energy condition) for the Einstein Equations. Recently, Mantoulidis and Schoen designed an innovative construction of asymptotically flat extensions, with non-negative scalar curvature, of a given 2-surface satisfying some geometric restrictions (related to the stability inequality for minimal surfaces). This extension has the feature that its asymptotic behavior is well controlled. In particular, they were able to calculate the Bartnik mass, in the minimal case, of the given surface. In this talk, the adaptation of this construction to other settings of interest in mathematical relativity will be discussed; in particular, its application to obtain Bartnik mass estimates for CMC Bartnik data. This talk is based on joint projects with A. Alaee, C. Cederbaum, S. McCormick and P. Miao.

•  Supporting material in the links below: