Starts 12 May 2023 14:15
Ends 12 May 2023 16:15
Central European Time
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IGAP COMPLEX DIFFERENTIAL GEOMETRY SEMINARS
IGAP (former Sissa Building)
Lecture room 205
via Beirut 2
14:00 -15:00 - Reza Seyyedali (IPM, Teheran, Iran)
Some a priori estimate for constant scalar curvature Kahler metrics
Abstract: The problem of existence of constant scalar curvature Kahler metrics (cscK) on compact Kahler manifolds has been studied for many years. In the case of complex one dimension, the answer is provided by the uniformization theorem. It states that any compact Riemann surface admits a metric of constant Gaussian curvature. One way to prove this fact is to solve a semi linear elliptic equation. In higher dimensions, cscK metrics satisfy a fully nonlinear fourth order elliptic equation. It is extremely difficult to deal with such equations partially due to lack of maximum principle. In a recent breakthrough, Chen and Cheng proved some important a priori estimates for cscK metrics. More generally they showed that in a given Kahler class, the set of metrics with uniform bounded scalar curvature and bounded relative entropy is compact in C^{\infty} topology. In a joint work with Z. Lu, we generalized their results. Mainly, we replace the uniform boundedness assumption on scalar curvature with some L^p boundedness.
15:15 - 16:15 - Romina M. Arroyo (FAMAF & CIEM - Universidad Nacional de Cordoba, Argentina)
Invariant SKT structures on nilmanifolds
Abstract: A very active research field in complex geometry arises from imposing Hermitian metrics with special properties. One of the most studied cases is the one of "strong Kähler with torsion" (SKT for short) geometry. In this talk, we will introduce SKT Hermitian manifolds, with a particular emphasis on invariant SKT structures on Lie groups. After that, we will discuss the existence of invariant SKT structures on nilmanifolds. More precisely, we will prove that an invariant SKT structure on a nilmanifold forces the nilmanifold to be at most 2-step nilpotent and we will provide examples of invariant SKT structures on 2-step nilmanifolds in arbitrary dimensions. This talk is based on a joint work with Marina Nicolini.
All are welcome to attend
Some a priori estimate for constant scalar curvature Kahler metrics
Abstract: The problem of existence of constant scalar curvature Kahler metrics (cscK) on compact Kahler manifolds has been studied for many years. In the case of complex one dimension, the answer is provided by the uniformization theorem. It states that any compact Riemann surface admits a metric of constant Gaussian curvature. One way to prove this fact is to solve a semi linear elliptic equation. In higher dimensions, cscK metrics satisfy a fully nonlinear fourth order elliptic equation. It is extremely difficult to deal with such equations partially due to lack of maximum principle. In a recent breakthrough, Chen and Cheng proved some important a priori estimates for cscK metrics. More generally they showed that in a given Kahler class, the set of metrics with uniform bounded scalar curvature and bounded relative entropy is compact in C^{\infty} topology. In a joint work with Z. Lu, we generalized their results. Mainly, we replace the uniform boundedness assumption on scalar curvature with some L^p boundedness.
15:15 - 16:15 - Romina M. Arroyo (FAMAF & CIEM - Universidad Nacional de Cordoba, Argentina)
Invariant SKT structures on nilmanifolds
Abstract: A very active research field in complex geometry arises from imposing Hermitian metrics with special properties. One of the most studied cases is the one of "strong Kähler with torsion" (SKT for short) geometry. In this talk, we will introduce SKT Hermitian manifolds, with a particular emphasis on invariant SKT structures on Lie groups. After that, we will discuss the existence of invariant SKT structures on nilmanifolds. More precisely, we will prove that an invariant SKT structure on a nilmanifold forces the nilmanifold to be at most 2-step nilpotent and we will provide examples of invariant SKT structures on 2-step nilmanifolds in arbitrary dimensions. This talk is based on a joint work with Marina Nicolini.
All are welcome to attend