UCSD Differential Geometry Seminar 2022

Unless otherwise noted, all seminars are on Thurs 11:00 - 12:00 am in Room 7321. Zoom id: 949 1413 1783


Winter 2022 Schedule

January 13

Alessandro Audrito, ETH
Title:A rigidity result for a class of elliptic semilinear one-phase problems
Abstract:
We study minimizers of a family of functionals arising in combustion theory, which converge, for infinitesimal values of the parameter, to minimizers of the \emph{one-phase free boundary} problem. We prove a $C^{1,\alpha}$ estimate for the ``interfaces'' of \emph{critical points} (i.e. the level sets separating the {\em burnt} and {\em unburnt} regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\mathbb{R}^N$ for $N \le 4$, answering positively a conjecture of Fern\'andez-Real and Ros-Oton. Our results are to the one-phase free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces. This is a joint work with J. Serra (ETHZ).

January 20

Davide Parisi
Title: Convergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n - 2)-area functional.
Abstract
We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern. The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. After starting with a general overview of the codimension one theory, we will move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied in gauge theory. We will explain to what extent the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interpret the former as a relaxation/regularization of the latter. We will mention some elements of the proof, with special emphasis on the role played by the gradient flow.

January 27

Gunhee Cho, UCSB
Title: The lower bound of the integrated Carath ́eodory-Reiffen metric and Invariant metrics on complete noncompact Kaehler manifolds.
Abstract
We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete K ̈ahler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carath ́eodory-Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base K ̈ahler metric with the Bergman metric, the Kobayashi-Royden metric, and the complete Ka ̈hler-Einstein metric in the conjecture class but missing of the Carath ́eodory-Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carath ́eodory-Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric, we establish the equivalence of the Bergman metric, the Kobayashi-Royden metric, and the complete Ka ̈hler-Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric on an n-dimensional complete noncompact Ka ̈hler manifold with some reasonable conditions which also imply non-vanishing Carath ́edoroy-Reiffen metric. This is a joint work with Kyu-Hwan Lee.

February 3

Xiaolong Li, Whichita
Title:Curvature operator of the second kind and proof of Nishikawa's conjecture
Abstract:
In 1986, Nishikawa conjectured that a closed Riemannian manifold with positive curvature operator of the second kind is diffeomorphic to a spherical space form and a closed Riemannian manifold with nonnegative curvature operator of the second kind is diffeomorphic to a Riemannian locally symmetric space. Recently, the positive case of Nishikawa's conjecture was proved by Cao-Gursky-Tran and the nonnegative case was settled by myself. In this talk, I will first talk about curvature operators of the second kind and then present a proof of Nishikawa's conjecture under weaker assumptions.

February 10


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February 17

Guido De Philippis
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February 24

Karoly Boroczky, UC Davis
Title: Stability of the log-Minkowski problem in the case of hyperplane symmetries
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March 3



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March 10


Andrea Marchese
Title: Tangent bundles for Radon measures and applications
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A powerful tool to study the geometry of Radon measures is the decomposability bundle, which I introduced with Alberti in [On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, GAFA, 2016]. This is a map which, roughly speaking, captures at almost every point the tangential directions to the Lipschitz curves along which the measure can be disintegrated. In this talk I will discuss some recent applications of this flexible tool, including a characterization of rectifiable measures as those measures for which Lipschitz functions admit a Lusin type approximation with functions of class C^1, the converse of Pansu's theorem on the differentiability of Lipschitz functions between Carnot groups, and a characterization of Federer-Fleming flat chains with finite mass.


Spring 2021 Schedule

Fall 2022 Schedule

Questions: lni@math.ucsd.edu