Conference On Geometric Analysis

Transcription

1 Conference On Geometric Analysis May, 2018, Tsinghua University DEPARTMENT OF MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY YAU MATHEMATICAL SCIENCES CENTER Organising Committee Shiu-Yuen Cheng (Tsinghua) Ernst Kuwert (Freiburg) Yuxiang Li (Tsinghua) Jiaping Wang (Minnesota) Guoyi Xu (Tsinghua)

2 Schedule for Conference on Geometric Analysis May, 2018 Beijing Time 21 May, Monday 22 May, Tuesday 23 May, Wednesday 24 May, Thursday 25 May, Friday Chair Shiu-Yuen Cheng Ernst Kuwert Yuxiang Li Franz Pedit Jiaping Wang 9:00-10: Tobias Lamm Richard Bamler Robert Haslhofer 10:45-12:00 Shing-Tung Yau Xiping Zhu Jingyi Chen Miles Simon Ovidiu Munteanu 12:00-12:20 Group photo 12:00-14:00 Lunch Chair Hui Ma Zuoqin Wang Bobo Hua -- 14:00-15:15 Lei Ni Bing-Long Chen Felix Schulze -- 15:45-17:00 Yuguang Shi Weimin Sheng Otis Chodosh -- 18:00-20:00 Banquet *Locations: Conference: Zheng Yu-Tong Lecture Hall of New Science Building Accommodation: Jiasuo Hotel Lunch: TBA Banquet Site: TBA

3 1. Richard Bamler, University of California, Berkeley Title: Classification of diffeomorphism groups of 3-manifolds through singular Ricci flow Abstract: I will present recent work of Bruce Kleiner and myself in which we classify the homotopy type of all spherical and hyperbolic 3-manifolds, with the exception of RP 3 in the spherical case. This partially resolves the Generalized Smale Conjecture in the spherical case and reproves a theorem due to Gabai in the hyperbolic case. Our proof is based on a new uniqueness theorem for singular Ricci flows, which we have established in previous work. Singular Ricci flows were introduced by Kleiner and Lott and are similar to Perelmans Ricci flows with surgery, as used in his resolution of the Poincar and Geometrization Conjectures. In contrast to Perelmans surgery process, which is carried out at a positive scale and depends on a number of auxiliary parameters, a singular Ricci flow is more canonical, as it flows through surgeries at an infinitesimal scale. Our uniqueness theorem allows the study of continuous families of singular Ricci flows, providing important information on the diffeomorphism group of the underlying manifold. 2. Bing-Long Chen, Sun Yat-Sen University Title: An isoperimetric inequality on discrete groups and its geometric applications Abstract: In this talk, a refined isoperimetric inequality on discrete groups will be introduced. We prove that automatic groups and CAT(0) groups satisfy such an inequality for degree=2. We will describe some applications in Kahler geometry. This is a joint work with Xiao Kui Yang. 3. Jingyi Chen, University of British Columbia Title: Some recent progress on Hamiltonian stationary Lagrangian submanifolds Abstract: We will talk about some recent advances on Hamiltonian stationary Lagrangian submanifolds in C n. These are critical points of the volume functional of Lagrangian submanifolds under Hamiltonian variations, and they satisfy a fourth order elliptic equation locally. We will discuss the regularity and compactness problems based on joint work with M. Warren, and geometry of Hamiltonian stationary cones in joint work with Y. Yuan as well. 4. Otis Chodosh, Princeton University Title: Min-max applications of the Allen-Cahn equation on 3- manifolds 1

4 Abstract: I will describe recent joint work with C. Mantoulidis in which we prove curvature, multiplicity, and index estimates for minimal surfaces arising from the Allen Cahn min-max construction. In particular, these results imply that a generic Riemannian 3-manifold has the property that for every positive integer p, there is an embedded, two-sided, smooth minimal surface whose Morse index is precisely equal to p. 5. Robert Haslhofer, Toronto University Title: Minimal two-spheres in three-spheres Abstract:We prove that any manifold diffeomorphic to S 3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover. 6. Ovidiu Munteanu, University of Connecticut Title: Structure of four dimensional shrinking Ricci solitons Abstract: I will describe recent results about the asymptotic geometry of complete four dimensional shrinking Ricci solitons. These are self similar solutions to the Ricci flow and appear in blowups of singularities of the flow. Combined with work by Brett Kotschwar and Lu Wang, we can formulate an approach for the classification of complete four dimensional shrinkers. This talk is based on joint work with Jiaping Wang 7. Tobias Lamm, Karlsruher Institut fr Technologie Title: Expanders of the harmonic map heat flow Abstract: Expanding self-similarities of a given evolution equation create an ambiguity in the continuation of the flow after it reached a first singularity. In this talk, we investigate the possibility of smoothing out any map from the n-sphere, n > 1, to any closed Riemannian manifold, that is homotopic to a constant by a self-similarity of the harmonic map flow. We also study the singular set of such solutions as well as the uniqueness issue. This is a joint work with Alix Deruelle. 8. Lei Ni, University of California, San Diego 2

5 Title: Metric characterizations of the projectivity Abstract: Recently there were several progresses on the algebraic and analytic properties of Kaehler manifolds in terms of holomorphic sectional curvature. In this talk I shall discuss some new joint results (with Fangyang Zheng) along these directions. The result either generalizes the existing one or provides a complementary coverage. The new curvature conditions also opens many questions. 9. Felix Schulze, University College London Title: Optimal isoperimetric inequalities for 2-dimensional surfaces in Hadamard-Cartan manifolds in any codimension Abstract: Let (M n, g) be simply connected, complete, with non-positive sectional curvatures, and Σ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that S = Σ. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from Σ, to show that S satisfies the optimal Euclidean isoperimetric inequality: 6 πm[s] (M[Σ]) 3 2. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by κ < 0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in L Weimin Sheng, Zhejiang University Title: A class of anisotropic curvature type flows and the Aleksandrov and dual Minkowski problems Abstract: In this talk I will introduce our recent work on a contracting flow of closed, convex hypersurfaces in the Euclidean space R n+1 with speed of fr α σ k, where σ k is the k-th elementary symmetric polynomial of the principal curvatures, α R, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α k + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f 1. When k = n, our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-minkowski problem introduced by Huang, Lutwak, Yang and Zhang. If α < k +1, counterexample is given for the above smooth convergence. This is our joint work with Qi-rui Li and Xu-jia Wang. 11. Yuguang Shi, Peking University 3

6 Title: Quasi-local mass and uniqueness of isoperimetric surfaces in asymptotically hyperbolic manifolds Abstract: Quasi-local mass is a basic notion in General Relativity. Geometrically, it can be regarded as a geometric quantity of a boundary of a 3-dimensional compact Riemannian manifold. Usually, it is in terms of area and mean curvature of the boundary. It is interesting to see that some of quasi-local masses, like Brown-York mass, Hawking mass and isoperimetric mass have deep relations with classical isoperimetric inequality in asymptotically flat (hyperbolic) manifolds. In this talk, I will discuss these relations and finally give an application in the uniqueness of isoperimetric surfaces in asymptotically Ads-Schwarzschilds manifold with scalar curvature. This talk is based on my recent joint works with M.Echmair, O.Chodosh and my Ph.D student J. Zhu. 12. Miles Simon, Otto von Guericke University Title: Local Ricci flow and limits of non-collapsed regions whose Ricci curvature is bounded from below Abstract: We use a local Ricci flow to obtain a bi-hlder correspondence between noncollapsed (possibly non-complete) 3-manifolds with Ricci curvature bounded from below and Gromov-Hausdorff limits of sequences thereof. This is joint work with Peter Topping and the proofs build on results and ideas from recent papers of Hochard and Topping+Simon. 13. Shing-Tung Yau, Harvard University Title: Quasi-local mass at null innity 14. Xi-Ping Zhu, Sun Yat-Sen University Title: Regularity of Harmonic Maps between Singular Spaces Abstract: M. Gromov and R. Schoen in 1992 initiated to study the theory of harmonic maps into singular spaces. In 1997, J. Jost and F. H. Lin, independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. Meanwhile, F. H. Lin proposed an open question: can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces. In this talk I will present an affirmative answer to it. Moreover, I will discuss how to get quantitative gradient estimates in term of a lower bound of Ricci 4

7 curvature. This is based on joint works with Hui-Chun Zhang and Xiao Zhong. 5