Abstract: Building upon the Almgren's big regularity paper, Chang proved in the eighties that the singularities of area-minimizing integral 2-dimensional currents are isolated. His proof relies on a suitable improvement of Almgren's center manifold and its construction is only sketched. In recent joint works with Emanuele Spadaro and Luca Spolaor we give a complete proof of the existence of the center manifold needed by Chang and extend his theorem to two classes of currents which are "almost area minimizing", namely spherical cross sections of area-minimizing 3-dimensional cones and semicalibrated currents.
Abstract: An initial data set in a spacetime M is a triple (V,h,K), where V is a spacelike hypersurface in M, h is its induced (Riemannian) metric and K is its second fundamental form. A solution to the Einstein equations for physically relevant sources influences the curvature of V via the Einstein constraint equations. In this talk we will discuss some aspects of the geometry and topology of initial data sets, all of which involve or have been greatly influenced by Rick's work. In particular, we will consider the topology of black holes in higher dimensional gravity, inspired by certain developments in string theory and issues related to black hole uniqueness. We will also discuss recent work on the geometry and topology of the region of space exterior to all black holes, which is closely connected to the notion of topological censorship. Many of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.
Abstract: Various classes of chains and cochains may reveal geometric as well as topological properties of metric spaces. In 1957, Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space, completed to get flat chains, and defined flat cochains as the topological dual space. Federer and Fleming also considered these in the sixties and seventies, for homology and cohomology of Euclidean Lipschitz neighborhood retracts. These sets include smooth manifolds and polyhedra, but not algebraic varieties or subspaces of some Banach spaces. In works with Thierry De Pauw and Washek Pfeffer, we find generalizations and alternate variational topologies for flat chains and cochains in general metric spaces. With these, we homologically characterize Lipschitz path connectedness and obtain several facts about and examples of spaces that satisfy local linear isoperimetric inequalities.
Abstract: The talk will introduce a family of folded hyperkahler structures on a disc bundle in the cotangent bundle of a compact surface and discuss a possible generalisation of Teichmuller space as a moduli space of Finsler structures defined by the boundary circle bundle.
Abstract: We prove that it is possible to get families of catenoids as limit leaves of a limit lamination of embedded minimal disks. The construction works for certain rotationally invariant riemannian metrics on three-manifolds, including hyperbolic space. Our method allows us to give another counterexample to the general Calabi-Yau conjecture for hyperbolic space, producing a complete and embedded - but not properly embedded - simply connected minimal surface on either side of any area-minimizing catenoid in hyperbolic space.
Abstract: In this talk I will discuss joint work with Joaquin Perez, Antonio Ros, Giuseppe Tinaglia and Pablo Mira. Joint work with Perez and Ros (and Harold Rosenberg) has led to the completion of the classification of properly embedded minimal surfaces of genus 0; these examples are planes, catenoids, helicoids and Riemann minimal examples. Joint work with Tinaglia proves that compact disks of constant mean curvature 1 embedded in R3 have curvature estimates away from their boundaries and that there exists a universal bound on the intrinsic radius of such disks. Consequently, any complete, simply-connected embedded surface in R3 with non zero constant mean curvature must be a round sphere. This work with Tinaglia also implies that a complete embedded surface in R3 of positive constant mean curvature has bounded second fundamental forms if and only if it has positive injectivity radius. We also prove that a complete embedded surface of constant mean curvature in R3 is proper if it has finite topology or positive injectivity radius. Joint work with Perez and Ros gives removable singularity results for constant mean curvature laminations and these results lead to an better understanding of the local structure of CMC foliations of 3-manifolds near any isolated singularity. My talk ends with an outline of my recent proof with Mira, Perez and Ros of the Hopf Uniqueness Theorem in homogenous 3-manifolds. This generalization shows that two oriented immersed spheres of the same mean curvature in a homogeneous 3-manifold X are congruent and provides a description of the associated 1-dimensional moduli space M(X) is parameterized by the mean curvature. When $X$ is topologically the three-sphere, then M(X) is naturally analytically parameterized by the real numbers and the "unique" minimal sphere in X is embedded and has an extrinsic isometry group that contains the dihedral group D(4).
Abstract: I will present a construction of entire solutions of some nonlinear elliptic equations (such as the nonlinear Schrodinger equation, the magnetic Ginzburg-Landau equations of the Chern-Simons-Higgs equations) which have no symmetry. This extends the construction of N. Kapouleas on the construction of compact constant mean curvature surfaces in euclidean 3-space.
Abstract: We present some results related to a question posed by Berestycki, Caffarelli and Nirenberg. We study the geometry of non-compact smooth planar domains Ω bounded by finitely many curves and admitting a bounded positive solution u:Ω → R of the overdetermined problem
∆u + f(u) = 0 in Ω
u=0 and ∂u/∂n = 1 on ∂Ω
where n is the inward pointing unit normal vector and f:[0,∞) → R is a given Lipschitz function. This is a joint work with D. Ruiz and P. Sicbaldi.
Abstract: Joint work with Stephan Tillmann. We define a generalisation of the recent construction of Gay and Kirby of trisections of 4-manifolds to multisections of smooth or PL n-manifolds. These are decompositions into k handlebodies, where n=2k or 2k+1. The intersections of the handlebodies have spines with substantial codimension and the intersection of all the handlebodies are closed submanifolds with Cat(0) structures.
Abstract: The talk will briefly survey known results and open questions related to cylindrical minimal cones and their significance to questions such as the structure of the singular set of minimal submanifolds and the growth properties of entire solutions of the minimal surface equation.
Abstract: In joint work with Martin Sack we show that the solutions to the Cauchy problem for a wave equation with critical exponential nonlinearity in 2 space dimensions scatter for arbitrary smooth, compactly supported initial data.
The positive mass theorem, a greatest accomplishment in the theory of general relativity, paves the way for a deep understanding of the notion of mass. Many important tools in geometric analysis such as minimal surfaces and the conformal method were brought in to study this fundamental yet subtle notion. Another important geometric PDE, the isometric embedding equation, arose naturally in the study of quasi-local mass recently. I shall review these new developments, and in particular discuss how the rigidity property of mass (when is mass equal to zero) is intimately related to the uniqueness of the isometric embedding problem. This talk is based on joint work with Po-Ning Chen and Shing-Tung Yau.