Introductory Workshop: Special Geometric Structures and Analysis: Geometric flows of G_2 and Spin(7)-structures
Presenter
September 5, 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
- minimal surfaces
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
We will discuss a family of flows of G_2-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G_2-structures. We will prove short-time existence and uniqueness of solutions to the flows and a priori estimates for some specific flows in the family. We will discuss analogous flows of Spin(7)-structures. This talk is based on arXiv:2311.05516 (joint work with P. Gianniotis and S. Karigiannis) and arXiv:2404.00870.