5th Geometry Day

Schedule

• 13:00 - 13:25 : Jinwoo Shin (Sookmyung Women's University)

• 13:30 - 13:55 : Yunhee Euh (Sungkyunkwan University)

• 14:00 - 14:25 : Seunghoon Jeong (Postech)

• 14:25 - 15:00 : (Break time)

• 15:00 - 15:25 : Jihyeon Lee (IBS-CGP) 

• 15:30 - 15:55 : Jungwoo Moon (Chung-Ang University)

• 16:00 - 16:25 : Sanghun Lee (Pusan National University)

• 16:30 - 16:55 : Eungmo Nam (Korea Institute for Advanced Study)

Title & Abstract

• Jinwoo Shin (Sookmyung Women's University) 

Title : Equivariant Yamabe-type problems 

Abstract: Since its resolution in the 1980s, the Yamabe problem has inspired many generalizations and related geometric questions.  Among them, the equivariant Yamabe problem studies the existence of constant scalar curvature metrics that respect a prescribed group symmetry. 

In this talk, we review the formulation of the equivariant Yamabe problem and discuss several results concerning the existence of equivariant solutions, together with some recent developments in this direction.

• Yunhee Euh (Sungkyunkwan University) 

Title : Critical Metrics for the Quadratic Riemann Curvature Functional on Complete Manifolds

Abstract: We investigate critical metrics of the quadratic Riemann curvature functional $\mathcal{A}(g)=\int_M |R|^2\, dV_g$ on complete Riemannian manifolds $(M^n,g)$ with finite energy, i.e. $\mathcal{A}(g)<\infty$. In dimension four, we show that every complete $\mathcal{A}$-critical metric is either Einstein or locally isometric to a Riemannian product of two $2$-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$, with $c\neq 0$. Our result extends to the complete noncompact case the known compact classification of four-dimensional $\mathcal{A}$-critical metrics. 

• Seunghoon Jeong (Postech)

Title : Asymptotic convergence of nonconvex Wasserstein gradient flows 

Abstract: In 2001, Otto introduced a weak Riemannian structure on the $L^2$-Wasserstein space of probability measures with finite second moment. Since then, gradient flows in the optimal transport framework have been extensively studied due to their broad range of applications. However, a general theory for nonconvex energy functionals remains largely open. 

In this talk, I will briefly review the geometric interpretation of optimal transport, known as the Otto calculus. I will then discuss the asymptotic convergence of Wasserstein gradient flows for certain nonconvex functionals by applying the Łojasiewicz–Simon theory on the Wasserstein tangent space. This is joint work with Beomjun Choi (KAIST) and Geuntaek Seo (POSTECH).

• Jihyeon Lee (IBS-CGP)

Title : Hamiltonian Stationary Twisted Lagrangian Tori in $\mathbb{C}^n$

Abstract : Chekanov's exotic tori play an important role in symplectic geometry as examples of Lagrangian tori that are not Hamiltonian isotopic to the standard product torus. A useful class of such Lagrangian tori can be constructed by twisting a simple closed planar curve, producing what are known as twisted Lagrangian tori. In a recent work, Chen and Coulibaly studied the Hamiltonian stationary condition for twisted Lagrangian tori in $\mathbb{C}^2$ and proved that the only Hamiltonian stationary twisted tori arise from circles centered at the origin, i.e., the Clifford product torus. 

In this talk, we generalize their framework to higher-dimensional complex space $\mathbb{C}^n$, and derive explicit geometric formulas for Hamiltonian stationary equation for twisted tori generated by planar curves. In particular, we show that the Hamiltonian stationary condition imposes strong restrictions on the generating curve and that, under natural assumptions, the only Hamiltonian stationary twisted tori in $\mathbb{C}^n$ are the product tori. These results provide a higher-dimensional perspective on the geometry of twisted Lagrangian tori and clarify the role of Chekanov-type constructions in Hamiltonian stationary geometry. 

• Jungwoo Moon (Chung-Ang University) 

Title : Some properties on symmetric tautness tensor 

Abstract : In this talk, we recall the definition of symmetric $2$-tensor on a Riemannian manifold with a Riemannian foliation. After the remind, some geometric properties of Riemannian foliations are derived by applying symmetric tautness tensor. 

• Sanghun Lee (Pusan National University) 

Title : Some results on weighted marginally outer trapped surfaces with free boundary

Abstract: In this talk, we introduce the notion of weighted marginally outer trapped surfaces (WMOTS) and explore geometric results for WMOTS with free boundary. First, under a suitable dominant energy condition, we prove that WMOTS with free boundary have positive m-Bakry-\'{E}mery scalar curvature and weighted minimal boundary. Second, we estimate the area of WMOTS and analyze their topology and geometric properties.  

•  Eungmo Nam (Korea Institute for Advanced Study)

Title : The ancient pancake solution of the mean curvature flow 

Abstract: In this talk, we will study the ancient pancake solution of the mean curvature flow constructed by Bourni–Langford–Tinaglia, provides a detailed analysis of its geometric and asymptotic properties, and discusses the problem of characterizing its uniqueness under a small entropy condition.