Asymptotically hyperbolic (AH) manifolds arise naturally in general relativity as a spacelike hypersurface in an asymptotically Minkowski spacetime or an asymptotically AdS spacetime. The mass of AH manifolds is a geometric invariant that measures its deviation from hyperbolic space. In this talk, we present a new mass formula using large coordinate horospheres. The formula is stated as a limit of the (weighted) total difference of mean curvatures on large coordinate horospheres. We remark on a few geometric implications of the formula including scalar curvature rigidity of AH manifolds. This talk is based on joint work with Pengzi Miao.
Non-uniqueness for the anisotropic Calderon problem.
The anisotropic Calderon problem consists in recovering, up to some natural gauge equivalences, the metric of a compact Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map for the Laplacian, at fixed energy. The Calderon problem has been the object of a significant amount of research activity in geometric analysis since it was first formulated by Calderon in 1980, and is still open in its most general form. After giving a motivated introduction to the problem, we shall review its current status and present some recently obtained counter-examples to uniqueness. The latter results, obtained in collaboration with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes), involve an unexpected mixture of conformal geometry and classical analysis.
On three-manifolds admitting a Killing vector field.
A Killing submersion is a Riemannian submersion, from an orientable three-manifold to an orientable surface, whose fibers are the integral curves of a Killing vector field. Indeed, any three-manifold with a Killing vector field can be locally endowed with a Killing submersion structure away from zeroes of the vector field. The first part of this talk will be devoted to discuss local and global classification results for Killing submersions by prescribing an arbitrary base surface and two arbitrary geometric functions, namely the bundle curvature and the length of the Killing vector field. In this discussion, we will assume that the total space is either Riemannian or Lorentzian. Moreover, we will show that the only Killing submersions with Killing vector field of constant length in which the total space is homogeneous are the Bianchi-Cartan-Vranceanu spaces, i.e., the so-called E(κ,τ)-spaces. Time permitting, we will also discuss a geometric duality for graphical spacelike surfaces between Riemannian and Lorentzian Killing submersions that swaps the mean curvature of the surface and the bundle curvature of the ambient space. This talk is partially based on joint works with Hojoo Lee and Ana M. Lerma.
Vanishing and Estimation results for Betti numbers.
We prove that manifolds with ⌈n/2⌉-positive curvature operators are rational homology spheres. This is a consequence of a general vanishing and estimation theorem for the p-th Betti number for manifolds with a lower bound on the average of the lowest (n−p) eigenvalues of the curvature operator. Our main tool is the Bochner Technique. We will also discuss similar results for the Hodge numbers of Kaehler manifolds. This talk is based on joint work with Peter Petersen.
Conformally Kähler Einstein-Maxwell metrics on blow-ups.
Conformally Kähler Hermitian metrics of constant Riemannian scalar curvature and J-invariant Ricci are called conformally Kähler Einstein-Maxwell metrics. In this talk, we discuss deformations and possible constructions of such metrics on blow-ups. This is a joint work in progress with Abdellah Lahdili.
Geometric flows have emerged as important tools in the application of analysis to geometry. The most basic example of a geometric flow is the curve shortening flow. At its simplest, this is a flow of plane curves, in which a plane curve is deformed in the direction of its normal vector with speed given by the curvature. It has been studied extensively in R2, especially when the flowing curve is closed. In this talk, we study the curve-shortening flow in hyperbolic 2-space. We will prove a general theorem characterizing the curves that evolve in a self-similar manner. Analogous results for curves in the 2-sphere have recently been obtained by Santos dos Reis and Tenenblat (2019) and for curves in the plane by Halldorsson (2012). Time permitting, we will discuss curves that flow so as to remain self-similar up to an overall scaling, including curves that extend to infinity and have a well-defined renormalized length. This is joint work with Ran Xie and should be accessible to undergraduates who have had a first course in curves and surfaces.
J-holomorphic curves or pseudoholomorphic curves have been of great interest to mathematics since the discovery that J-holomorphic curves allow to construct invariants of symplectic manifolds by Gromov and that those invariants are related to topological superstring theory by Witten. A crucial step towards Gromov–Witten invariants is the compactification of the moduli space of J-holomorphic curves via stable maps which was first proposed by Kontsevich and Manin. In this talk, I want to report on a supergeometric generalization of J-holomorphic curves and stable maps where the domain is a super Riemann surface. Super Riemann surfaces have first appeared as generalizations of Riemann surfaces with anti-commutative variables in superstring theory. Super J-holomorphic curves and super stable maps couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and might ultimatively lead to a supergeometric generalization of Gromov-Witten invariants. Based on arXiv:2010.15634 & arXiv:1911.05607, joint with Artan Sheshmani and Shing-Tung Yau.
There has recently been a great deal of activity on Scalar Curvature, both on the index theoretic and metric geometry side. The classic work of Gromov-Lawson and Schoen-Yau is a backdrop for the more modern developments, and here we will describe some recent work with Unger-Yau which seeks to generalize some of the classic results to the setting of non-compact manifolds. Connections with problems in Conformal Geometry, Positive Mass Theorems will be alluded to, and other current developments by other authors will be described. This is based on arXiv:2009.12618.
Geometric flows have been one of the most fruitful research areas in geometric analysis the past decades with vast applications such as the resolution of the Geometrization conjecture, the Riemannian Penrose Inequality and the differentiable sphere theorem. Often, the main goal here is to precisely understand the long-time behavior of these geometric flows. In 1996 Stahl studied free boundary mean curvature flow (i.e. mean curvature flow with Neumann boundary data) whose barrier surface is given by a round sphere and showed that convex surfaces converge to shrinking hemispheres along the flow. This can be compared to Huisken’s celebrated convergence result for mean curvature flow. An alternative approach to Stahl’s result has been given by Edelen in 2016. This raises the natural question whether the assumption of the barrier being a round sphere can be removed and that a similar result holds true for arbitrary convex barrier surfaces. Using a novel 5-tensor perturbation technique we answer this question affirmatively in the 2-dimensional setting. This work is joint with Martin Li.
The Einstein-Dirac equation I: The conformal case.
In this first talk I will introduce the Einstein-Dirac functional and its Euler Lagrange equation and then I will focus on the conformal version of the functional. I will show the bubbling phenomena and the energy quantization that occurs in this model and provide an existence result under an Aubin type condition. Next, I will state one recent result classifying the ground state bubbles that appear during blow-up. (The first part of the talk is a joint work with V. Martino and the second is a joint work with W. Borrelli.)