Schedule-at-a-Glance

14 PRIMA 2013 AbstractsRicci flow and 4-manifolds with positiveisotropic curvatureHong HuangBeijing Normal University, Chinahhuang@bnu.edu.cnWe’ll survey the work by Hamilton, Chen, Tang, Zhu andthe speaker on the classification of complete 4-manifolds(or orbifolds) with positive isotropic curvature. The maintool used here is the Hamilton-Perelman theory on Ricciflow with surgery. When we consider noncompact manifoldsor orbifolds we need to adapt the original Hamilton-Perelman theory. We’ll indicate the necessary modificationsin these cases.Convergence of Calabi flow with small initialdataHaozhao LiUniversity of Science and Technology of China, Chinahzli@ustc.edu.cnWe will discuss the long time existence and convergence ofthe Calabi flow under some small initial conditions withoutassuming the existence of constant scalar curvatureKähler metrics. This is joint work with Kai Zheng.K-stability of Fano varieties and Alpha invariantYuji OdakaKyoto university, Japanyodaka@math.kyoto-u.ac.jpThe K-stability is first defined by Tian and later generalizedby Donaldson formally as a positivity of generalizedFutaki invariants. This talk will focus on the case of Fanovarieties.Thanks to the recent celebrated works of the proofof existence of Kähler-Einstein metrics on K-stable Fanomanifolds due to Chen-Donaldson-Sun and Tian, the K-stability has been finally proved to be equivalent to theexistence of Kähler-Einstein metrics i.e. we can in principlestudy the existence problem algebro-geometrically.However it is in general hard, at the moment, to testK-stability.The speaker will review the basic structure of the generalizedFutaki invariants from algebro-geometric viewpointand gives relation with (the Minimal modelprogram-based) birational geometry, in particular introducingthe notion of “destabilizing subschemes" afterthe wake of Ross-Thomas. This analysis in particularwill give a proof of K-stability of Fano n-fold X withα(X) >n which corresponds to the theorem of Tiann+1in 80s where the alpha invariant α(X) was introduced.This talk will be based on a joint work with Yuji Sano.α-functions of smooth del Pezzo surfacesJihun ParkInstitute for Basic Science & Pohang University of Scienceand Technology, Koreawlog@postech.ac.krWe define α-functions of Fano varieties by considering theα-invariants of G. Tian locally. We demonstrate how toobtain the α-functions of smooth del Pezzo surfaces. Inaddition, their applications are briefly introduced.Bergman kernel of a polarized Kähler ALEmanifoldYalong ShiNanjing University, Chinashiyl@nju.edu.cnI shall report some recent joint work with Claudio Arezzoand Alberto Della Vedova on the Bergman kernel of KählerALE manifolds.The first eigenvalue of minimal submanifoldsin an unit sphereZizhou TangBeijing Normal University, ChinaWe will talk about the first eigenvalue of a minimalisoparametric hypersurface in a unit spheres, as well asthat of their focal submanifolds. For the special case, weverify Yau’s conjecture.The regularity of limit spaceBing WangUniversity of Wisconsin-Madison, USAbwang@math.wisc.eduThis is a joint work with Tian. We study the structure ofthe limit space of a sequence of almost Einstein manifolds,which are generalizations of Einstein manifolds.Roughlyspeaking, such manifolds are the initial manifolds of somenormalized Ricci flows whose scalar curvatures are almostconstants over space-time in the L 1 -sense, Ricci curvaturesare bounded from below at the initial time. Underthe non-collapsed condition, we show that the limitspace of a sequence of almost Einstein manifolds has mostproperties which is known for the limit space of Einsteinmanifolds. As applications, we can apply our structureresults to study the properties of Kähler manifolds.The limit of the Yang-Mills-Higgs flow onsemi-stable Higgs bundlesJiayu Li & Xi ZhangUniversity of Science and Technology of China, Chinamathzx@ustc.edu.cnIn this talk, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitianvector bundle (E, H 0 ) over a compact Kähler manifold(M, ω). We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show thatthe limiting Higgs sheaf is isomorphic to the double dualof the graded Higgs sheaves associated to the Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle.Ricci curvature in Kahler-Ricci flowZhou ZhangUniversity of Sydney, Australiazhangou@maths.usyd.edu.auRicci curvature is a geometric quantity naturally relatedto the behaviour of Ricci flow. In this talk, we discusssome recent results on the relations between the variousbounds of Ricci curvature and the Kähler-Ricci flow existingfor either finite or infinite time.A class of Weingarten curvature measuresBin ZhouPeking University, Chinabzhou@pku.edu.cn