2026 Mini-workshop on Riemannian and Complex Geometry

등록 및 discussion

Title: ``Mean Curvature'' of the Bergman Metric

Speaker: 염지훈(전북대학교)

Abstract: In the first session, we will introduce the fundamental concepts of the Bergman kernel and Bergman metric, covering their essential properties and their pivotal role in Complex Geometry. we will also discuss recent work, where we investigate ``Statistical Bergman Geometry''—a novel approach that examines Bergman geometry from the perspective of Information Geometry.

Title: ``Mean Curvature'' of the Bergman Metric

Speaker: 염지훈(전북대학교)

Abstract: In the second session, we will introduce a new invariant: the ``mean curvature'' of the Bergman metric. As this research is in its early stages, the main goal of this presentation is to share preliminary findings with experts in mean curvature and to seek practical advice and various constructive feedback for future development.

Title: TBA

Speaker: 표준철(부산대학교)

Abstract: 

Title: On Stationary Real Matrix Schubert Varieties

Speaker: 이재훈(KIAS)

Abstract: Many known area-minimizing cones are realized as real algebraic varieties. Since algebraic varieties are defined by explicit polynomial equations, they provide clear geometric intuition through concrete computations. Recent results have shown that all determinantal varieties are indeed minimal submanifolds. In this talk, we discuss a necessary condition for real matrix Schubert varieties, a natural generalization of determinantal varieties within the class of algebraic varieties, to be minimal. The results presented in this talk are based on joint work with Sangwoo Park and Eungbeom Yeon.

Title: Symmetric Differentials of Holomorphic Ball Bundles

Speaker: 이승재(경북대학교)

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 $E$ whose fiber is the complex unit ball $\mathbb{B}^n$, and the structure group is the 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 $B^n$ bundle can be embedded into the associated $\mathbb{P}^n$ bundle $\widehat E$ as a relatively compact smooth domain.

Suppose that $\mathbb{B}^n$ bundle has a holomorphic section $s:M \rightarrow E$ which is an embedding. Then $M$ can be identified with a compact complex submanifold of $E$. When the conormal bundle of $M$ in $E$ is ample, it becomes an interesting problem to study implicit/explicit global global $L^2$ extension for symmetric differentials of $M$, and to ask relations between weighted $L^2$ holomorphic functions on $E \subset \widehat E$ and symmetric differentials on $M$.

In the case of when $M$ is a compact hyperbolic space form, this problem has been investigated by several authors, including M. Adachi, S. Lee, A. Seo. In this talk, I will explain these line of researches and give some proofs if time permits.

Title: Gromov K{\"a}hler Hyperbolicity and Eigenvalue Estimates on Bounded Symmetric Domains

Speaker: 조예원(경상국립대학교)

Abstract: In 1991, Gromov introduced the notion of Kähler hyperbolic manifolds which in particular generalizes Kähler manifolds of Riemannian sectional curvature bounded from above by a negative constant. Gromov's basic estimate on such manifolds yields a vanishing theorem for harmonic forms and also a lower bound for the eigenvalues of the Laplacian of the given Kähler metric. The bound is determined by a uniform constant and the so-called `Kähler hyperbolicity length' of the metric.

In this talk, I shall explain a method to obtain lower bounds for the eigenvalues of the Laplacian of the complete Kähler-Einstein metrics of Ricci curvature -1 on bounded symmetric domains, using the aforementioned estimate. The method in particular provides the optimal lower bound on the complex hyperbolic space (and polydiscs) which is sharper than McKean's estimate (1970). This is joint work with Young-Jun Choi and Kang-Hyurk Lee.

3일 차 - 2026.01.07

Title: TBA

Speaker: 표준철(부산대학교)

Abstract: 

Title: Local Rigidity of Three-Manifolds via the Magnetically Charged Hawking Mass

Speaker: 이지현 (IBS-CGP)

Abstract: We study compact minimal surfaces in the Einstein--Maxwell theory with both electric and magnetic charges and a negative cosmological constant. Working in the time-symmetric setting and under the magnetically charged dominant energy condition, we show that any two-sided, embedded, strictly stable minimal surface that maximizes the magnetically charged Hawking mass naturally corresponds to the event horizon of a black hole. Our main rigidity theorem proves that the ambient geometry near such a surface is uniquely determined: a neighborhood is isometric to the dyonic Reissner--Nordstr\"{o}m--Anti--de Sitter model, the canonical charged black hole in AdS spacetime. Moreover, we obtain an area estimate depending only on the surface topology and the relevant physical parameters (charges and cosmological constant). These results give a clean mathematical characterization of black-hole horizons in the charged AdS regime via a variational property of a modified Hawking mass. This is a joint work with Sanghun Lee.