| Date/Time/Venue | Talks |
|---|---|
| Jan 23, 2019 (Wed) 11:00-12:00 @ AB1 502a |
(no seminar, see MIST 2019 Workshop) Speaker: Zuoqin Wang (USTC) Title: Equivariant Eigenvalues for Manifolds with Large Symmetry Abstract: Consider a Riemannian manifold M whose Riemannian metric is invariant under the action of a compact Lie group G. Then the Laplacian eigenvalues decompose naturally according to the irreducible representations of $G$. I will explain the role of symplectic geometry in studying these equivariant eigenvalues. In particular, I will show how to apply symplectic techniques to study inverse spectral problems on toric manifolds. This is based on joint works with V. Guillemin. |
| Feb 20, 2019 (Wed) 11:00-12:00 @ AB1 502a |
Speaker: Eungbeom Yeon (Seoul National University) Title: Characterizations of the plane and the catenoid as capillary surfaces Abstract: In this talk, I will prove that a capillary minimal surface outside the unit ball in R^3 with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. |
| Feb 27, 2019 (Wed) 14:00-15:00 @ AB1 502a |
(MIST Workshop on Geometric Analysis, Algebraic Geometry and Symplectic Geometry (GAS)) Speaker: Niels Martin Moller (University of Copenhagen) Title: The wedge theorem for ancient mean curvature flows Abstract: We show that a wedge theorem (also called a bi-halfspace theorem) holds for properly immersed ancient solutions to the mean curvature flow in R^(n+1). This adds to a long story, as it generalizes our own wedge theorem for self-translaters from 2018, which implies the minimal surface case by Hoffman-Meeks (1990) that in turn contains the classical cone theorem by Omori (1967). Another application of the wedge theorem is to classify the convex hulls of the sets of reach of all proper ancient flows, hence posing restrictions on the possible singularities than can occur in mean curvature flow. The proof uses a parabolic Omori-Yau maximum principle for proper ancient flows. This is joint work with Francesco Chini (U Copenhagen). |
| Feb 27, 2019 (Wed) 16:00-17:00 @ AB1 502a |
(MIST Workshop on Geometric Analysis, Algebraic Geometry and Symplectic Geometry (GAS)) Speaker: Jaigyoung Choe (KIAS, CUHK) Title: The isoperimetric inequality for minimal surfaces in R^n (Part I) Abstract: The domain with largest area for fixed circumference in plane is the disk. It has been conjectured that the minimal surface with largest area for fixed circumference in R^n is also the flat disk. I will survey the history of this open conjecture. In later seminars I will talk about the isoperimetric inequality for minimal surfaces in a Riemannian manifold (Part II) and about the joint work with R. Schoen in R^3 (Part III). |
| Mar 13, 2019 (Wed) 11:00-12:00 @ AB1 502a |
Speaker: Jaigyoung Choe (KIAS, CUHK) Title: The isoperimetric inequality for minimal surfaces in a Riemannian manifold (Part II) Abstract: First, it will be proved that a minimal surface S of fixed area and with 1 or 2 boundary components in hyperbolic space has the shortest boundary only when S is a totally geodesic disk. Then we will consider surfaces in a Riemannian manifold with curvature bounded from above by a nonpositive constant. |
| Mar 20, 2019 (Wed) 11:00-12:00 @ AB1 502a - CANCELLED - |
Speaker: Yannick Sire (Johns Hopkins University) - CANCELLED - Title: Minimal surfaces with free boundary, half-harmonic maps and Ginzburg-Landau approximation Abstract: I will report on recent results dealing with the construction of minimal surfaces with free boundary adopting the new objects known as half-harmonic maps introduced by Da Lio and Riviere. These maps are critical points of the H^{1/2} semi-norm on the real line. Their image appear to be minimal immersions with free boundary. I will also describe how to get these maps out of a Ginzburg-Landau regularization and raise several open problems and possible directions of research. |
| Mar 21, 2019 (Thur) 11:00-12:00 @ AB1 502a (Note special date and time.) |
Speaker: Alexander Grigor'yan (Bielefeld University) Title: On positive solutions of semi-linear elliptic inequalities on Riemannian manifolds Abstract: We are concerned with two problems about semi-linear elliptic inequalities: - existence of entire positive solutions - pointwise estimates of such solutions. The answers are given in terms of the volume function and Green function. |
| Mar 27, 2019 (Wed) 11:00-12:00 @ AB1 502a |
Speaker: Jaigyoung Choe (KIAS, CUHK) Title: The isoperimetric inequality for minimal surfaces in R^3 (Part III) Abstract: We will find some sufficient conditions for flat surfaces to satisfy the classical isoperimetric inequality. Then it will be shown that a minimal surface in R^3 gives rise to two flat surfaces. And we will study when the sharp isoperimetric inequality for the minimal surface follows from that of the two flat surfaces. (joint with R. Schoen) |
| Mar 28, 2019 (Thur) 11:00-12:00 @ AB1 502a (Note special date and time.) |
Speaker: Chao Xia (Xiamen University) Title: Stability on two types of partitioning problems Abstract: In this talk, two geometric variational problems, so-called partitioning problems for convex domains will be discussed. Type-I is on area minimizing hypersurfaces with volume constraint and Type-II is on area minimizing ones with wetting area constraint. The stationary points for Type-I are free boundary CMC hypersurfaces while the ones for Type-II are minimal hypersurfaces with constant contact angle. We study the stability problem, namely, the second variation for area functional is nonnegative under two types of admissible deformations. We show the uniqueness result for stable type-I or type-II stationary hypersurfaces in a Euclidean ball. This talk is based on joint works with Guofang Wang and Jinyu Guo. |
| Apr 10, 2019 (Wed) 11:00-12:00 @ AB1 502a |
(Part of MIST 2019 Workshop III) Speaker: Wei Yuan (Sun Yat-Sen University) Title: Gap phenomena for obstruction-flat metrics and its topological applications Abstract: Obstruction tensor is a trace-free symmetric 2-tensor, which indicates whether a given representative for the conformal infinity of a conformally compact Einstein manifold can be smoothly extended to the interior of the manifold. As a special case for 4-dimensional manifolds, obstruction tensor is exactly the well-known Bach tensor. In this talk, we will present a type of gap phenomena associated to closed manifolds with vanishing obstruction tensor. Moreover as an application, we will study the topology of 8-dimensional locally conformally flat manifolds and give an estimate of its Euler characteristic. This talk is based on a series of joint works with Dr. Fang Yi in Anhui University of Technology. |
| Apr 17, 2019 (Wed) 11:00-12:00 @ AB1 502a |
Speaker: Junbin Li (Sun Yat-Sen University) Title: On the theory of the formation of trapped surface and black hole Abstract: The study of the formation of trapped surface and black hole, and the large data problem in general relativity, is pioneered by Christodoulou in 2008, in his work "The formation of black hole in general relativity". In this talk, I will briefly introduce my works on the formation of trapped surface and black hole, including the construction of Cauchy data that leads to the formation of trapped surface, and some works on the weak cosmic censorship conjecture, which is one of the fundamental problems in general relativity. |
| Apr 24, 2019 (Wed) 11:00-12:00 @ AB1 502a |
Speaker: Kwok-Kun Kwong (University of Sydney) Title: Sharp Levy-Gromov type isoperimetric inequalities Abstract: The Levy-Gromov isoperimetric inequality states that under a positive Ricci curvature lower bound, the area-to-volume ratio of a domain is not smaller than that of a certain ball in the comparison space. In this talk, I will present a new sharp Levy-Gromov type isoperimetric inequality which involves the cut distance. There are two new features. One is that we allow the Ricci curvature lower bound to be arbitrary. The second one, which is perhaps more surprising, is that we obtain a lower bound for the volume instead of an upper bound (of course the bound cannot depend only on the boundary area, but also on its cut distance). This contrasts with the classical isoperimetric inequality, the Levy-Gromov isoperimetric inequality and the Bishop-Gromov volume comparison theorem, all of which give an upper bound of the volume of a domain either in terms of its boundary area, or in terms of the volume of its counterpart in the comparison space. If time allows, I will also talk about another isoperimetric inequality which involves the extrinsic radius of a domain. |