| Date/Time/Venue | Talks |
|---|---|
| Jan 23, 2018 (Tue) 16:00-17:00 @ AB1 502a |
Speaker: Ling Yang (Fudan University) Title: On Dirichlet problem for minimal graphs and Lawson-Osserman constructions Abstract: We develop the Lawson-Osserman's works on minimal graphs. Firstly, we construct a constellation of uncountably many Lawson-Osserman spheres, which are minimal in Euclidean spheres and therefore generate Lawson-Osserman cones that correspond to Lipschitz but non-differentiable solutions to the minimal surface system. Then, by the theory of autonomous systems in plane, we find for each Lawson-Osserman cone an entire minimal graph having it as tangent cone at infinity. Further, in addition to the truncated Lawson-Osserman cones, we discover infinitely many analytic solutions to the Dirichlet problem of minimal surfaces system for boundary data induced by certain Lawson-Osserman spheres. As a corollary, those Lawson-Osserman cones are non-minimizing. These behaviors are observed for the first time. This is the joint work with Prof. Xiaowei Xu and Yongsheng Zhang. |
| Jan 25, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Joon Tae Kim (Seoul National University) Title: Wrapped Floer homology of real Lagrangians and volume growth of fibered twists Abstract: Fibered twists are certain symplectomorphisms that can be defined on a Liouville domain whose boundary has a periodic Reeb flow. We investigate an entropy-type invariant, called the slow volume growth, of the component of fibered twists and give a uniform lower bound of the growth using wrapped Floer homology. We apply our results to examples from real symplectic manifolds. They admit so-called real Lagrangians, and we compute wrapped Floer homology using Morse-Bott techniques. This is joint work with Myeonggi Kwon and Junyoung Lee. (a part of MIST 2018 seminar series) |
| Jan 26, 2018 (Fri) 10:30-11:30 @ AB1 501a |
Speaker: Yanyan Niu (Capital Normal University) Title: Gap theorem on Kahler manifold Abstract: In this talk, we will talk about a gap theorem for Kahler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature. This is a joint work with Professor Lei Ni. |
| Feb 1, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Xiang Ma (Peking University) Title: Pseudo-convex submanifolds — curvature properties, Plateau problem, and the discrete theory Abstract: As a generalization of the usual convex hypersurfaces, we consider a spacelike submanifold immersed in a pseudo Euclidean space, whose normal bundle is induced with a Lorentz inner product of signature (1,p), and we make a convexity assumption on its second fundamental form. The simplest example is a closed curve in the 3-dimensional Lorentz space whose tangent and normal vectors span a spacelike plane, and it winds around some timelike axis with index 1. For such a pseudo-convex loop, we show that it satisfies a reversed Fenchel type inequality (its total curvature is no more than 2pi), and it always span a spacelike maximal surface. Then we give the general definition of a pseudo-convex submanifold and demonstrate their close relationship with convex hypersurfaces. If time is allowed we will briefly mention what is a pseudo-convex polyhedron and a discretized Positive Mass Conjecture about the Liu-Yau mass. This is a joint work with Dr. Nan Ye, and the final part is an ongoing project with my postdoc Dong Zhang. |
| Feb 5, 2018 (Mon) 11:00-12:00 @ AB1 502a |
Speaker: Mao-Pei Tsui (National Taiwan University) Title: Curvature estimates in higher codimensional mean curvature flow Abstract: K. Ecker and G. Huisken have derived a priori estimate for the curvature (second fundamental forms) when they study the mean curvature flow of the graph of a function in Euclidean space. In this talk, I will explain that a similar curvature estimate also exists for higher codimensional mean curvature flow under certain natural conditions. (part of MIST 2018: Symposium on Geometric Analysis) |
| Feb 27, 2018 (Tue) 16:00-17:00 @ Tse Chiu Kit Room |
Speaker: Simon Brendle (Columbia University) Title: Curvature and topology of manifolds Abstract: The interplay between curvature and topology of Riemannian manifolds is among the most fundamental questions in differential geometry. Over the past century, various different approaches have been developed to attack these types of problems. This includes variational techniques based on geodesics and minimal surfaces, as well as the Ricci flow approach pioneered by Richard Hamilton. In this lecture, I will give an overview of the subject, focusing on the positive curvature setting. (Joint Colloquium-part of MIST 2018 program) |
| Mar 15, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Yoshihiro Tonegawa (Tokyo Institute of Technology) Title: Existence and regularity for the Brakke flow Abstract: The Brakke flow is a generalized mean curvature flow in the framework of geometric measure theory. The notion is general enough to include motion of singular objects such as networks and soap bubble clusters while it is equipped with a powerful regularity theory. I mainly explain the content of the recent existence theorem and give an outline of how to construct the time-discrete approximate flows. Ref. (1) L. Kim, Y. Tonegawa, On the mean curvature flow of grain boundaries, Ann. l’Institut Fourier (Grenoble) 67, (2017) 43-142, (2) K. Kasai, Y. Tonegawa, A general regularity theory for weak mean curvature flow, Calc. Var. PDEs 50, (2014) 1-68. |
| Mar 16, 2018 (Fri) 10:30-11:30 @ AB1 502a |
Speaker: Yoshihiro Tonegawa (Tokyo Institute of Technology) Title: Various results for singular perturbation problems of diffused interface Abstract: The Modica-Mortola (or Allen-Cahn) energy has been widely used in mathematical modeling to represent the hypersurface area of thin diffused interface region. The usefulness does not stop at the modeling, on the other hand. The energy is equipped with a rich hidden structure and one can establish sharp and rigorous results in the framework of geometric measure theory on its singular perturbation limit under various assumptions. I give an overview on results in this direction. Ref. (1) Y.Tonegawa, N. Wickramasekera, Stable phase interfaces in the van der Waals-Cahn-Hilliard theory, J. Reine Angew. Math. 668 (2012), 191-210, (2)K. Takasao, Y. Tonegawa, Existence and regularity of mean curvature flow with transport term in higher dimensions, Math. Ann. 364, (2016), 857-935,(3) Y. Tonegawa, A diffused interface whose chemical potential lies in Sobolev spaces, Ann. Sc. Norm. Sup. Pisa 4 (2005) 487-510. |
| Mar 22, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Zhichao Wang (Peking University) Title: Free boundary minimal hypersurfaces with least area Abstract: In this talk, I will present a joint work with Q. Guang and X. Zhou where we prove the existence of least area free boundary minimal hypersurfaces in compact manifolds with boundary. Moreover, we characterize the orientation and index of such minimal hypersurface. As a consequence, it is exactly the min-max one corresponding to the fundamental class if the ambient manifold has non-negative Ricci curvature and convex boundary. |
| Mar 29, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Wenlong Wang (Peking University & IMS) Title: Rigidity of Riemannian Penrose inequality for asymptotically flat 3-manifolds with corners Abstract: In this talk, we will talk about a rigidity result for the equality case of the Penrose inequality on asymptotically flat 3-manifolds with nonnegative scalar curvature and corners. This result is closely related to the boundary behavior of compact manifolds with nonnegative scalar curvature. This is joint work with Yuguang Shi and Haobin Yu. |
Apr 19, 2018 (Thur) 10:30-11:30 @ AB1 502a |
Speaker: Chung-Jun Tsai (National Taiwan University) Title: Rigidity of the minimal sphere in the Atiyah-Hitchin manifold Abstract: In a hyper-Kahler 4-manifold, holomorphic curves are stable minimal surfaces. One may wonder whether those are all the stable minimal surfaces. Micallef gave an affirmative answer in many cases. However, this cannot be true in general. Micallef and Wolfson proved that the minimal sphere in the Atiyah-Hitchin manifold is strictly stable, but cannot be holomorphic with respect to any compatible complex structure. They conjectured that the minimal sphere in the Atiyah-Hitchin manifold is indeed quite rigid. In this talk, we will first review the construction of the Atiyah-Hitchin manifold, and then explain the rigidity of that minimal sphere. This is based on a joint work with Mu-Tao Wang. |