The seminar usually takes place in 515, Cosmology Building at 4-5PM every Wednesday, unless noted otherwise.
Organizers: Siao-Hao Guo 郭孝豪 (NTU), Chao-Ming Lin 林朝明 (NTU), Wei-Bo Su 蘇瑋栢 (NCU), Chung-Jun Tsai 蔡忠潤 (NTU)
The Second Workshop on Geometric Variational Problems, Geometric Measure Theory, and Related Topics
Junyoung Park 박 준영 (Rutgers University): Ancient mean curvature flow asymptotic to a minimal quadratic cone
Mean curvature flow (henceforth MCF) is one of the most extensively geometric flows that evolves hypersurfaces in Euclidean space by its mean curvature vector. One of the central topics in MCF theory is the classification of ancient solutions, possibly with an extra geometric assumption. In this talk, we discuss the rigidity of ancient MCF that is asymptotic to a minimal quadratic cone in high dimensions. We first show that if such flow lies on one side of the cone, then it has unique asymptotics. We then show that under an additional mean convexity assumption, such flow has to be a stationary solution given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity / unique asymptotics result for ancient MCF asymptotic to a singular model shrinker.
Tang-Kai Lee 李堂愷 (Columbia University)
Tsz Kiu Aaron Chow 周子翹 (The Hong Kong University of Science and Technology)
Brian Harvie (University of Copenhagen)
Kai-Wei Zhao 趙凱衞 (University of California, Riverside): Obata-type rigidity on static manifolds with boundary
Static manifolds naturally arise in general relativity and as obstructions to deformation of scalar curvature. In this talk, we will study rigidity and deformability properties of b-static metrics on simple manifolds with compact boundary, imposing a Robin-type boundary condition on static potentials, associated with the coupled map of scalar curvature and boundary mean curvature. By establishing a spectral correspondence between the induced boundary static operator and the stability operator of marginally outer trapped surfaces (MOTS), we show that boundary stability enforces global positivity of b-static potentials, thereby excluding the existence of interior horizons. Building on this spectral framework, we obtain an Obata-type rigidity classification by using an integral identity. We further derive sufficient geometric conditions ensuring the local surjectivity of the coupled curvature map. This talk is based on joint work with Hongyi Sheng.
Guoran Ye 叶果然 (Northwestern University): Special Lagrangians with Cylindrical Tangent Cones
We construct special Lagrangian submanifolds $Y\subset \C^{n+1},n\geq 3$ in a neighborhood of the origin, with an isolated singularity, but with cylindrical tangent cone $C\times\R$. Such examples exist, for example, when $C$ is a pair of transverse planes. The construction glues exact Lagrangian graphs. While the construction is purely analytic, it is important to recognize that the leading term of the Lagrangian potential in the gluing region exhibits cohomological information.
Yi-Lin Tsai 蔡宜霖 (NCTS): Constant Rank Theorems in the Complex Settings
Constant rank theorems are useful tools in geometric analysis for studying degeneracy and rigidity in nonlinear partial differential equations.
In this talk, we will discuss a quantitative constant rank theorem for positive semidefinite $(1,1)$-tensors on Hermitian manifolds, extending the result of Sz\'{e}kelyhidi and Weinkove from the real setting.
We will explain the positivity and tensor conditions needed in this more general setting, together with the main ideas of the proof. As an application, we study the rank of the Ricci tensor on certain Hermitian manifolds. This part of the talk is based on previous work. If time permits, we will also discuss some recent developments in constant rank theorems and related questions.