Mini-workshop on Riemannian and complex geometry

1일 차 - 2023.07.10

Problems on the uniformization of negatively curved K\"ahler manifolds

강연자: 이강혁

Abstract:
The uniformization theorem of the Riemann surfaces says that a simply connected Riemann surface is conformally equivalent to Riemann sphere $\mathbb{C}\mathbb{P}^1$ or the complex plane $\mathbb{C}$ or the unit disc $\Delta$. This gives the complete list of universal covering spaces of Riemann surfaces. Classical theories such as the Schwarz lemma and the Gauss-Bonnet theorem imply that most Riemann surfaces can be covered by the unit disc. This crucial feature is based on the negative curvature of the Poincar\'e metric of $\Delta$. In Several Complex Variables, theories on the Poincar\'e metric have been developed under name of the geometric function theory. However, there are still many problems to be solved for the multi-dimensional uniformization. A fundamental problem is to classify simply connected complex manifolds covering a negatively curved compact K\"ahler manifold. In this talk, I will discuss relevant problems and my research program, especially the potential rescaling method for the existence of complete holomorphic vector fields.

Lecture 1: First and Second variation of minimal surfaces

강연자: 서검교

Abstract:
The first variation of a minimal surface refers to the change in area resulting from small deformations of the surface. For a minimal surface, the first variation of the area is zero, which means that the surface is stationary with respect to small deformations. The second variation of a minimal surface measures how the surface's area changes to second order when subjected to variations. If the second variation is positive for all nontrivial variations, the surface is said to be stable. This means that the minimal surface is a local minimum of area. On the other hand, if the second variation is negative for some variations, the surface is unstable, indicating a local maximum or a saddle point. We discuss the first and second variation of minimal surfaces.

Complete embedded minimal surfaces in the doubled Schwarzschild 3-manifold

강연자: 이재훈

Abstract:
Complete embedded minimal surfaces of finite total curvature in three-dimensional Euclidean space can either have bounded height or have logarithmic growth. Schoen initially discovered this fact using the Weierstrass representation, and more recently, Bernard and Rivi\`ere extended this result to asymptotically flat spaces. We provide a comprehensive review of these findings and explore the existence of complete embedded minimal surfaces that correspond to the two aforementioned asymptotic properties within the doubled Schwarzschild 3-manifold-an important example of an asymptotically flat space. This talk is based on the joint work with Choe and Yeon.

Statistical Bergman Geometry

강연자: 염지훈

Abstract:

For a bounded domain $\O$ in $\CC^n$, let $P(\O)$ be the set of all (real) probability distributions on $\O$. Then, in general, $P(\O)$ becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric, called the {\it Fisher information metric}, on $P(\O)$. Information geometry studies a finite-dimensional submanifold $M$, which is called a {\it statistical manifold}, in $P(\O)$ using geometric concepts such as Riemannian metric, distance, connection, and curvature, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods.

In this talk, we first introduce a map $\Phi : \O \rightarrow P(\O)$ and prove that the pull-back of the Fisher information metric on $P(\O)$ is exactly same as the Bergman metric of $\O$.

This map provides a completely new perspective that allows us to view Bergman geometry from a statistical viewpoint. We will discuss several results in this framework.

This is a joint work with Gunhee Cho at UC Santa Barbara University. 

New minimal surfaces in the doubled Schwarzschild 3-manifold

강연자: 연응범

Abstract : 

There are few examples of minimal surfaces in the Riemannian Schwarzschild manifolds in general. We draw our attention to the doubled Schwarzschild manifolds and seek for new minimal surfaces. In particular, we construct complete embedded minimal surfaces of finite topology via classical desingularization method.

Weighted L2 holomorphic functions on holomorphic ball bundles  

강연자: 이승재

Abstract:

In this talk, we will discuss relations between holomorphic functions and underlying geometric structure of certain complex ball bundles over complex Kähler manifolds. A holomorphic ball bundle over a complex manifold $M$ is a fiber bundle whose fiber is the complex unit ball $\mathbb{B}^n$, and the structure group is the automorphism group of all holomorphic automorphisms of $\mathbb{B}^n$. Since the automorphism group of $\mathbb{B}^n$ is canonically embedded into the automorphism group of $\mathbb{P}^n$, any ball bundle can be embedded into the associated $\mathbb{P}^n$ bundle as a relatively compact smooth domain.

In 2017, M. Adachi constructed weighted $L^2$ holomorphic functions on certain disc-fiber bundles over compact Riemann surfaces. In 2020, A. Seo and S. Lee extended his result to certain ball bundles over compact complex hyperbolic space forms using symmetric differentials. Recently, A. Seo and S. Lee showed that these phenomena are still valid for some compact submanifolds of finite volume ball quotients. In this talk, we will explain these results and give some proofs if time permits.

2일 차 - 2023.07.11

Lecture 2: Stable minimal surfaces

강연자: 서검교

Abstract:
The Jacobi operator, also known as the stability operator or the second variation operator, is a linear differential operator that characterizes the second variation of the area functional for a minimal surface. Studying the eigenvalues and eigenfunctions of the Jacobi operator allows us to analyze the stability and behavior of minimal surfaces. If all the eigenvalues of the Jacobi operator are positive, then the minimal surface is stable. This implies that any small variation of the surface will lead to an increase in the area to second order. In the lecture, we discuss some properties of stable minimal surfaces.

Problems on the uniformization of negatively curved K\"ahler manifolds

강연자: 이강혁

Abstract:
The uniformization theorem of the Riemann surfaces says that a simply connected Riemann surface is conformally equivalent to Riemann sphere $\mathbb{C}\mathbb{P}^1$ or the complex plane $\mathbb{C}$ or the unit disc $\Delta$. This gives the complete list of universal covering spaces of Riemann surfaces. Classical theories such as the Schwarz lemma and the Gauss-Bonnet theorem imply that most Riemann surfaces can be covered by the unit disc. This crucial feature is based on the negative curvature of the Poincar\'e metric of $\Delta$. In Several Complex Variables, theories on the Poincar\'e metric have been developed under name of the geometric function theory. However, there are still many problems to be solved for the multi-dimensional uniformization. A fundamental problem is to classify simply connected complex manifolds covering a negatively curved compact K\"ahler manifold. In this talk, I will discuss relevant problems and my research program, especially the potential rescaling method for the existence of complete holomorphic vector fields.