The seminar usually takes place in 515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Nicolau Sarquis Aiex 艾尼克 (NTNU), Siao-Hao Guo 郭孝豪 (NTU), Chung-Jun Tsai 蔡忠潤 (NTU)
Chun-Sheng Hsueh 薛駿勝 (Berlin Mathematical School): Open Books in Higher Dimensions
We study open book decompositions on manifolds of dimension at least 4 using handle decompositions. We use handle moves to relate different open books on the same manifold and show that an open book with trivial monodromy can be alternatively constructed using a simple page and a non-trivial monodromy. As an application, we provide infinitely many examples in all dimensions at least 5 of diffeomorphic open books constructed using the same page but non-isotopic monodromy maps.
Siao-Hao Guo 郭孝豪 (NTU): Stability and singular set of the two-convex level set flow
The level set flow of a mean-convex closed hypersurface is stable off singularities, in the sense that the level set flow of the perturbed hypersurface would be close in the smooth topology to the original flow wherever the latter is regular. To study the behavior near singularities, we further assume that the initial hypersurface is two-convex and that the flow has finitely many singular times. In this case, the singular set of the flow would have finitely many connected components, each of which is either a point or a compact $C^{1}$ embedded curve. Then under additional conditions, we show that near each connected component of the singular set of the original flow, the perturbed flow would have "the same type" of singular set as that of the singular component.
Kai-Wei Zhao 趙凱衞 (University of Notre Dame): Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves
Curve shortening flow is, in the compact case, the gradient flow of arc-length functional. It is a one-dimensional case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions is a parabolic version of geometric Liouville-type theorem. The previous results technically rely on the assumption of convexity of the curves. In the ongoing project joint with Kyeongsu Choi, Dong-Hwi Seo, and Wei-Bo Su, we replace the convexity condition by the boundedness of entropy, a measure of geometric complexity introduced by Colding and Minicozzi. In this talk, we show that an ancient smooth curve shortening flow with finite entropy embedded in R^2 has a unique tangent flow at infinity. Furthermore, we show that flow with entropy less than 3 must be a shrinking circle, a static line, a paper clip, or a translating grim reaper and show that if entropy m is greater than or equal to 3, then its rescaled flows backwardly converge to a line with multiplicity m exponentially fast in any compact region and m is an integer.
Mu-Tao Wang 王慕道 (Columbia University): Some Recent Advances in Geometric Analysis
I will discuss some recent advances concerning the isoperimetric inequality and the positive mass theorem, topics that are closely related to Professor Simon Brendle’s lectures next week.
AS-NCTS-NTU Distinguished Lectures in Geometry by Simon Brendle (Columbia University)
Keomkyo Seo 서검교 (Sookmyung Women's University): Overdetermined boundary value problems in a Riemannian manifold
Serrin's overdetermined problem is a famous result in mathematics that deals with the uniqueness and symmetry of solutions to certain boundary value problems. It is called "overdetermined" because it has more boundary conditions than usually required to determine a solution, which leads to strong restrictions on the shape of the domain. In this talk, we discuss overdetermined boundary value problems in a Riemannian manifold and discuss a Serrin-type symmetry result to the solution to an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature and it will be discussed about an overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold.
Jintian Zhu 朱锦天 (Westlake University): Splitting theorem in Kahler geometry
In this talk, we focus on the relationship between the curvature and the distance on Riemannian manifolds. First we review classical results like the Bonnet-Myers theorem and the Cheeger-Gromoll splitting theorem. Then we introduce the mixed curvature on Kahler manifolds and present our recent splitting theorem for Kahler manifolds with nonnegative mixed curvature. If time permits, I'll mention possible extensions of our work in comparison to known results for Ricci curvature.
Ming Hsiao 蕭明 (NTU): Gap Theorem on locally conformally flat manifold
In this talk, I will present a gap theorem for locally conformally flat Riemannian manifolds with non-negative Ricci curvature. The proof involves constructing complete solutions to the Yamabe flow that exhibit instantaneous bounded curvature as they evolve. Using this, I show that if the curvature decays quickly enough in an integral sense, then the manifold must be flat. This result partially generalizes earlier work by Chen–Zhu and Ma. This is joint work with Man-Chun Lee.
Gongping Niu (The Chinese University of Hong Kong): Singular Isoperimetric Hypersurfaces and CMC Hypersurfaces under Perturbed Riemannian Metrics
It is known that isoperimetric hypersurfaces in a smooth, compact (n+1)-manifold are smooth except for a closed set of singularities with codimension at least 7. This is also an optimal estimate with singular examples constructed in [Niu24]. So in the specific case where n+1 = 8, there exist manifolds with singular isoperimetric hypersurfaces, prompting the question of whether all isoperimetric hypersurfaces are generically smooth. This talk will discuss the asymptotic rate of the induced Jacobi field near some isolated singular points and explore how it implies the removal of singularities for isoperimetric or constant mean curvature (CMC) hypersurfaces with perturbed Riemannian metrics.ith perturbed Riemannian metrics.
Wei-Hung Liao 廖偉宏 (National Yang Ming Chiao Tung University): Optimal Discrete Laplacians for Tetrahedral Meshes in $\mathbb{R}^3$: A Comparison of Primal and Dual Constructions
Discrete Laplace operators are fundamental tools in geometry processing, simulation, and modeling. For applications requiring robustness and accuracy, it is crucial that the discrete operators preserve essential properties of their continuous counterparts. In this work, we investigate discrete Laplacians on tetrahedral meshes embedded in $\mathbb{R}^3$, focusing on the structural differences between primal and dual constructions. We demonstrate that the discrete Laplacian derived from the dual construction satisfies the Euler-Lagrange equation for the Dirichlet energy, and can be decomposed into contributions from the tetrahedron's faces and a discrete mean curvature term related to the surrounding space. Through mathematical analysis, we show that the dual construction yields a more optimal and structurally faithful Laplacian compared to the primal approach. Furthermore, our results reveal that the dual mean curvature is more responsive to mesh variations, particularly in regions with high curvature changes, outperforming several existing discrete curvature models.
Kobe Marshall-Stevens (Johns Hopkins University): Gradient flow of phase transitions with fixed contact angle
The Allen-Cahn equation is closely related to the area functional on hypersurfaces and provides a means to investigate both its critical points (minimal hypersurfaces) and gradient flow (mean curvature flow). I will discuss various properties of the gradient flow of the Allen-Cahn equation with a fixed boundary contact angle condition, which is used to gain insight into an appropriate formulation for mean curvature flow with fixed boundary contact angle. This is based on joint work with M. Takada, Y. Tonegawa, and M. Workman.
Woongbae Park (University of Rochester): Evolution of the metric along with harmonic map flow
Since Eells-Sampson in '64, harmonic map flow has been studied widely and provided useful techniques in various fields in geometry. But its understanding is complicated due to the so-called "bubble phenomenon". Recently, there are several attempts to modify domain metric during harmonic map flow. Teichmuller harmonic map flow introduced coupled system of harmonic map flow and evolution of the metric which is L^2 gradient flow of Dirichlet energy. Ricci-harmonic map flow is a combination of harmonic map flow and Ricci flow. And finally conformal heat flow of harmonic maps is a pair of equations of harmonic map flow and conformal evolution of the metric. In this talk, I will present these systems and provide known results. In particular, I will compare them in terms of bubbling and explain how the additional metric change affects the bubble formation.
Kai Xu 徐恺 (Duke University): 3-Manifolds with positive scalar curvature and bounded geometry
In this talk we will work towards proving the following theorem: a contractible 3-manifold with positive scalar curvature and bounded geometry must be diffeomorphic to R^3. The proof involves running an innermost weak inverse mean curvature flow on the manifold. This talk is based on joint work with Otis Chodosh and Yi Lai.
Arunima Bhattacharya (University of North Carolina at Chapel Hill): Lagrangian mean curvature equations and flows
In this talk, we will introduce the special Lagrangian equation and the Lagrangian mean curvature flow. We will discuss interior Hessian estimates for shrinkers and expanders of the Lagrangian mean curvature flow, and further extend this result to a broader class of Lagrangian mean curvature type equations. We assume the Lagrangian phase to be hypercritical, which results in the convexity of the potential of the initial Lagrangian submanifold. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds.
Jui-Yun Hung 洪瑞鋆 (University of Notre Dame): A Parametrization of Solutions to the Yamabe Equation with Isolated Singularities
The Yamabe equation arises naturally in differential geometry, with its solutions corresponding to conformal metrics of constant scalar curvature. In this talk, we focus on solutions defined on a punctured ball in R^n that exhibit a non-removable singularity at the center. I will begin with a brief historical overview of the study of singular Yamabe equations and then present recent progress in this direction. In joint work with Professor Qing Han, we parametrize solutions to the Yamabe equation with non-removable isolated singularities using harmonic functions, or more precisely, sequences of spherical harmonics. Previous results have shown that such singular solutions can be formally represented by infinite series determined by these sequences. We prove that these formal series indeed converge to genuine solutions of the Yamabe equation with a non-removable isolated singularity. Moreover, these solutions have designated asymptotic behaviors determined by the sequences of spherical harmonics.
Chung-Ming Pan 潘仲銘 (Université du Québec à Montréal): Canonical metric problem for singular vector bundles on non-Kähler spaces
In this talk, I will begin by reviewing the definition of Gauduchon metrics and some progress on the correspondence between the existence of Hermite-Einstein metrics and the stability notion over singular bundles/spaces. I will then introduce a singular version of Gauduchon's theorem and discuss its application to the Hermite-Einstein problem for reflexive sheaves (which can be recognized as a singular holomorphic vector bundle) satisfying the slope-stability condition on non-Kähler normal varieties. Finally, I will explain one of the main technical points that lies in obtaining uniform Sobolev inequalities for perturbed hermitian metrics on a resolution of singularities.
