The seminar usually takes place in R515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Siao-Hao Guo 郭孝豪 (NTU), Ulrich Menne 孟悟理 (NTNU), Chung-Jun Tsai 蔡忠潤 (NTU)
Tang-Kai Lee 李堂愷 (Massachusetts Institute of Technology): Uniqueness of conical singularities for mean curvature flows
The uniqueness problem of blowups near a singularity is one of the most important problems in the study of various geometric flows. In this talk, I will first talk about some known results of this uniqueness problem for mean curvature flows, and discuss some recent progress when the singularities are conical. These are based on joint work with Xinrui Zhao.
Christian Scharrer (University of Bonn): Around three and a half worlds in zero days --- On the curvature cost of short world trips
Given a closed surface embedded into Euclidean space, its Willmore energy is defined as the integrated squared mean curvature. I'll present a sharp inequality that bounds the length of the shortest closed geodesic from below provided the surface has Willmore energy less than the energy of three and a half unit spheres.
Yat-Hin Marco Suen 孫逸軒 (IBS-CGP): Lagrangian multi-sections and their toric equivariant mirror
The SYZ conjecture suggests a folklore that ``Lagrangian multi-sections are mirror to holomorphic vector bundles". In this talk, we prove this folklore for Lagrangian multi-sections inside the cotangent bundle of a vector space, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We will also introduce the notion of tropical Lagrangian multi-sections and the Lagrangian realization problem. The latter asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic conditions prescribed by a given tropical Lagrangian multi-section. In dimension 2, we solve the realization problem for those tropical Lagrangian multi-sections that satisfy the so-called N-generic condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a simply connected Lagrangian multi-section. This is a joint work Yong-Geun Oh.
Yangyang Li 李阳垟 (The University of Chicago): Area ratio bounds and the existence of anisotropic minimal surfaces
Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. Due to the lack of a monotonicity formula, the regularity theory for its critical points, anisotropic minimal surfaces, is much more challenging than the area functional case. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface for elliptic integrands in closed 3-dimensional Riemannian manifolds via min-max construction. This confirms a conjecture by Allard [Invent. Math.,1983] in dimension 3. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.
Yuan Shyong Ooi 黃垣熊 (Pusan National University): Rigidity result of graphical mean curvature flow translating solution
Translating solution of mean curvature flow (MCF) is a self-similar solution that moves by translation under MCF. In the first part, I will survey results regarding translating surface which is graphical and translates in the graphical axis direction (or vertically). In the non-graphical setting, we can WLOG assume the translation direction to be vertical. However for graphical setting, vertical translating direction is equivalent to mean convexity condition whereas for non-vertical translating direction, we might lose mean convexity. In the second part, I will talk about my recent work on the rigidity results of graphical translator moving in non-vertical direction. This is a joint work with Pyo and Ma (preprint arXiv:2210.03707).
Long-Sin Li 李龍欣 (University of California, Irvine): Willmore Flow on Complete Surfaces
We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result by Kuwert and Schätzle in 2002 and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.
Hsin-Chuang Chou 周鑫壯 (National Taiwan Normal University): Integral Decompositions of Varifolds
In the theory of PDEs, one central element is the Poincaré inequality, a special case of which is the constancy theorem: if the derivative vanishes, then the function must be constant on each connected component. To extend the theory to varifolds, it turns out that even the concept of components is non-trivial; such a theory was established by Menne. The present work was motivated by the need for a different notion of components to extend the previous theory to a multiple-valued setting. In this talk, we will first give a quick introduction to varifolds which gives a measure-theoretical generalization of manifolds, then discuss the notions of indecomposability of varifolds, and finally, present the idea of the proof of the existence theorem of integral decompositions.
Pak-Yeung Chan 陳柏楊 (University of California, San Diego): Curvature and gap theorems of gradient Ricci solitons
Ricci flow deforms the Riemannian structure of a manifold in the direction of its Ricci curvature and tends to regularize the metric. This provides useful information about the underlying space. Ricci solitons are special solutions to the Ricci flow and arise naturally in the singularity analysis of the flow. We shall discuss some curvature and entropy gap theorems of gradient Ricci solitons. This talk is based on joint works with Yongjia Zhang and Zilu Ma, Eric Chen and Man-Chun Lee.
Jiawei Liu 刘佳伟 (Nanjing University of Science & Technology): Conical Kähler-Ricci flow and its related topics
In this talk, I will first talk about the existence, regularity and uniqueness of the conical Kähler-Ricci flow on compact Kähler manifold, and then the stability of this flow on Fano manifold and its applications. At last, I will talk about the related open problems.
Christopher Ling-Po Kuo 郭令波 (University of Southern California): Sheaf theoretic methods in symplectic geometry
Unlike Riemannian geometry, symplectic geometry is a soft theory in that it does not admit any local invariants such as curvature. Detecting global invariants is thus an important question for the theory. Instead of the more standard method of pseudo-holomorphic disks, I will discuss a topological/algebraic approach using microlocal sheaf theory. In particular, the wrapped Fukaya category, a version of Fukaya category for open symplectic manifolds, admits a purely sheaf theoretic description. I will explain this equivalence with a concrete example and, if time permits, present a new sheaf theoretic construction which has features resembling the Fukaya ones.
Keita Kunikawa 國川慶太 (Tokushima University): Liouville type theorem for harmonic maps of controlled growth
We show a Liouville type result for harmonic maps from a manifold with nonnegative Ricci curvature into positively curved target under the condition that the maps have some growth condition. Our result can be interpreted as an improved version of Choi's classical work. Moreover, Schoen-Uhlenbeck's example shows that our growth condition is almost sharp. The proof relies on Ecker-Huisken's curvature estimate for minimal hypersurfaces. This talk is based on a joint work with Yohei Sakurai.
Brain Harvie (National Center for Theoretical Sciences): A Rigidity Theorem for Asymptotically Flat Static Manifolds and its Applications
In general relativity, many physically and mathematically important questions concern the uniqueness of the Schwarzschild space. For example, Israel's black hole uniqueness theorem states that the Schwarzschild space is the only asymptotically flat static manifold with stable minimal boundary, and similar uniqueness questions for photon surfaces and for static metric extensions have recently generated great interest.
In this talk, I will present a new approach to these questions that is based on a recently-discovered Minkowski inequality for asymptotically flat static manifolds. We prove for that the equality is achieved only on coordinate spheres in the Schwarzschild space under natural boundary assumptions. We derive several new uniqueness theorems for Schwarzschild using this rigidity. Notably, we establish global uniqueness of static metric extensions for the Bartnik data induced by Schwarzschild coordinate spheres in all dimensions less than 8. This is based on joint work Ye-Kai Wang of NYCU.
@Astro-Math 440
Lan-Hsuan Huang 黃籃萱 (University of Connecticut): Positive mass theorem for asymptotically locally hyperbolic manifolds
The asymptotically (locally) hyperbolic manifold appears in a fundamental role in the AdS/CFT correspondence. It motivates further study on the properties of global invariants defined at conformal infinity, such as the energy-momentum vector. We will give a partial survey on positivity of the energy-momentum vector with particular emphasis on the equality case.
@Astro-Math 440
Damin Wu 吳大明 (University of Connecticut): Bergman metric on some negatively curved complete Kahler manifolds
We first briefly review the notion of Bergman metric with motivation. Then, we will present some recent development on the Bergman metric on certain negatively curved complete Kahler manifolds, with some unifying estimates of the Bergman metric using the bounded geometry.
Kai-Wei Zhao 趙凱衞 (University of Notre Dame): Ancient curve shortening flow of low entropy
Curve shortening flow is, in some sense, the gradient flow of arc-length functional. It is the simplest geometric flow and is a special case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions can be viewed as a parabolic analogue of Bernstein’s problem for minimal surfaces. The previous results technically reply on the assumption of convexity of the curves. In the joint project with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace it by the boundedness of entropy, which a measure of geometric complexity defined by Colding and Minicozzi. In this talk, we will focus on the ancient solutions with entropy at most 2, which is the limiting model of “fingers” and “tails” of solutions with higher finite entropy.
Shih-Fang Yeh 葉士房 (Michigan State University): Instability of Big Bang
In this talk, we will construct a family of Big Bang models and investigate the stability of the Big Bang. To be more specific, we firstly assume the homogeneity of the warped product spacetime to get some Big Bang models as background solutions that blow up at finite time, and then try to investigate the behavior of the solution after nonhomogeneous perturbation of initial data. It turns out, under certain conditions on the background and smallness of perturbation, shock will form before the Big Bang, which implies the instability of Big Bang.
