Introductory Workshop: Special Geometric Structures and Analysis: Geometric estimates along the Kähler-Ricci flow, part one
Presenter
September 3, 2024
Keywords:
- Kähler manifolds
- Kahler metrics
- Einstein metrics
- canonical metrics
- special holonomy
- Calabi-Yau
- geometric elliptic and parabolic PDEs
- Pluripotential Theory
- variational approach
- Monge-Ampère equation
- area minimizing currents
- semicalibrated currents
MSC:
- 32Q15 - Kähler manifolds
- 32Q20 - Kähler-Einstein manifolds
- 32Q25 - Calabi-Yau theory (complex-analytic aspects)
- 32Q57 - Classification theorems for complex manifolds
- 32U05 - Plurisubharmonic functions and generalizations
- 32W20 - Complex Monge-Ampère operators
- 35B65 - Smoothness and regularity of solutions to PDEs
- 35J47 - Second-order elliptic systems
- 49Q05 - Minimal surfaces and optimization
- 49Q15 - Geometric measure and integration theory
- integral and normal currents in optimization
- 49Q20 - Variational problems in a geometric measure-theoretic setting
- 53A10 - Minimal surfaces in differential geometry
- surfaces with prescribed mean curvature
- 53C07 - Special connections and metrics on vector bundles (Hermite-Einstein
- Yang-Mills)
- 53C38 - Calibrations and calibrated geometries
- 53C55 - Global differential geometry of Hermitian and Kählerian manifolds
Abstract
In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In these lectures, we will explain their approach, broaden the scope of their techniques, and apply these results to study the asymptotic behavior of the Kähler-Ricci flow.