Ye-Won Luke Cho (Gyeongsang National University)
TBA
Jiwon Jeong (Pusan National University)
Title: Infimum of the Spectrum of the Laplace–-Beltrami Operator on Domains in $\C^{n}$
Abstract: In this talk, we follow Li--Tran (2010) to study the infimum of the spectrum of the Laplace--Beltrami operator. After recalling that $\lambda_{1}(\mathbb{CH}^{n},\Delta_{g})=n^{2}$ on complex hyperbolic space $\mathbb{CH}^{n}$, we outline the argument that the same value holds for bounded strongly pseudoconvex domains. We then discuss bounded hyperconvex domains: the existence of a strictly plurisubharmonic exhaustion $\rho:D \rightarrow [-1,0)$ with finite Monge--Amp{\`e}re mass $\int_{D}(dd^{c}\rho)^{n} < +\infty$ allows one to adapt the same Rayleigh-quotient strategy and obtain $\lambda_{1}(D, \Delta_{g}) = n^{2}$. Finally, we discuss non-compact complete hyperconvex K{\"a}hler manifolds as a natural extension of the domain setting. Greene--Wu (1979) provides curvature-based examples of such manifolds, but these do not generally guarantee the existence of an exhaustion $\rho:M \rightarrow [-1,0)$ with finite Monge--Amp{\`e}re mass $\int_M (dd^c \rho)^n < +\infty$. We emphasize that finding such a finite-mass exhaustion would be the key step toward extending the same Rayleigh-quotient approach to the non-compact case.
Seunghoon Jeong (POSTECH)
Title: Riemannian structure on the L^2-Wasserstein space
Abstract: In 2001, Otto derived quantitative asymptotic results for the porous medium equation by describing a weak Riemannian structure on the $L^2$-Wasserstein space. Since this pioneering work, the geometry of $L^2$-Wasserstein spaces has been extensively developed and applied to the study of a wide range of partial differential equations by interpreting them as Wasserstein gradient flows.
In this talk, I will review the basic theory of optimal transport and prior studies establishing a Riemannian calculus on Wasserstein spaces. As an application, I will briefly discuss the work of Carrillo, McCann, and Villani on the granular media equation, as well as its recent generalizations by Choi, Jeong, and Seo.
Bowoo Kang (KAIST)
Title: Pluripotential solutions of complex Monge-Ampere flows
Abstract: In 1985, Cao reproved Yau's result on the Calabi conjecture using Kahler-Ricci flow. Since then, Kahler-Ricci flow has been an important topic in itself in the field of Kahler geometry. For its application to the analytic minimal model program proposed by Song and Tian, studying weak solutions of Kahler-Ricci flows under singular settings has been one of the main interests. In 2020, Guedj, Lu, and Zeriahi introduced a parabolic analogue of pluripotential theory and obtained results for pluripotential solutions of degenerate complex Monge-Ampere flows, which is a parabolic PDE equivalent to Kahler-Ricci flows. In this talk, I will present my recent works that generalize their results.
Jooho Lee (KIAS)
Title: Lagrangian Mean Curvature Flow in the Complex Euclidean Space
Abstract: Lagrangian mean curvature flow is a geometric evolution equation for Lagrangian submanifolds in Calabi-Yau manifolds, defined by moving the submanifolds in the direction of its mean curvature vector. Lagrangian mean curvature flow provides a natural link between differential geometry and partial differential equations. In this talk, we review the background and motivation for studying Lagrangian mean curvature flow.
Juncheol Pyo (Pusan National University)
Title: Eigenvalue of Laplacian and minimal submanifolds
Abstract: The Laplace–Beltrami operator is a central analytic object in geometry, and its eigenvalues encode striking information about the shape of a manifold and its submanifolds. Minimal submanifolds, critical points of the volume functional, are especially closely tied to harmonic and eigenfunction data. This talk surveys several classical and modern results illustrating how spectral information constrains and sometimes characterizes minimal geometry.
We begin with the basic fact about the eigenvalues of the Laplacian. We give the equivalence in Euclidean space: an isometric immersion is minimal if and only if its coordinate functions are harmonic. When the ambient space is the sphere, the coordinate functions satisfy an eigenvalue equation; this is a key feature of Takahashi’s theorem, linking minimal immersions into spheres with Laplacian eigenfunctions. Within this spectral viewpoint, we recall the Faber–Krahn theorem, which gives sharp lower bounds for the first Dirichlet eigenvalue in terms of volume and explains why balls optimize the fundamental tone. We then discuss the Choi–Wang theorem, providing a lower bound for the first Laplacian eigenvalue of compact embedded minimal hypersurfaces in ambient manifolds with positive Ricci curvature. Finally, we turn to free boundary minimal surfaces in the unit ball. In this setting, the relevant spectral object is the Steklov problem: under the orthogonality (free boundary) condition, the coordinate functions restrict to Steklov eigenfunctions, often realizing the first nontrivial Steklov eigenvalue. Together, these results show how Laplacian and Steklov spectra form a bridge between PDE and the geometric structure of minimal submanifolds.