Geometry and Topology Seminar: Difference between revisions
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Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show | Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show | ||
that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li. | that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li. | ||
===Ilyas Khan=== | |||
In this talk, we outline a proof of the following statement: Any mean curvature flow translator $\Sigma^2 \subset \R^3$ with finite total curvature and quadratic area growth must be a plane. | |||
== Archive of past Geometry seminars == | == Archive of past Geometry seminars == | ||
Revision as of 05:42, 10 November 2020
The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm. For more information, contact Shaosai Huang.
In the fall of 2020, we will hold online meetings on
Zoom platform
(available every Friday 1:00pm - 2:30pm).
Fall 2020
| date | speaker | title | host(s) |
|---|---|---|---|
| Oct. 23 | Yu Li (Stony Brook) | On the ancient solutions to the Ricci flow | (Huang) |
| Oct. 30 | Yi Lai (Berkeley) | A family of 3d steady gradient solitons that are flying wings | (Huang) |
| Nov. 6 | Jiyuan Han (Purdue) | On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations | (Chen) |
| Nov. 13 | Ilyas Khan (Madison) | Translating Surfaces with Finite Total Curvature are Planes | (Local) |
| Nov. 20 | Max Hallgren (Cornell) | TBA | (Huang) |
| Dec. 4 | Yang Li (IAS) | TBA | (Chen) |
Fall Abstracts
Yu Li
Ancient solutions model the singularity formation of the Ricci flow. In two and three dimensions, we currently have complete classifications for κ-noncollapsed ancient solutions, while the higher dimensional problem remains open. This talk will survey some recent developments of κ-noncollapsed ancient solutions with nonnegative curvature in higher dimensions.
Yi Lai
We found a family of $\mathbb{Z}_2\times O(2)$-symmetric 3d steady gradient Ricci solitons. We show that these solitons are all flying wings. This confirms a conjecture of Hamilton.
Jiyuan Han
Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li.
Ilyas Khan
In this talk, we outline a proof of the following statement: Any mean curvature flow translator $\Sigma^2 \subset \R^3$ with finite total curvature and quadratic area growth must be a plane.
Archive of past Geometry seminars
2019-2020 Geometry_and_Topology_Seminar_2019-2020
2018-2019 Geometry_and_Topology_Seminar_2018-2019
2017-2018 Geometry_and_Topology_Seminar_2017-2018
2016-2017 Geometry_and_Topology_Seminar_2016-2017
2015-2016: Geometry_and_Topology_Seminar_2015-2016
2014-2015: Geometry_and_Topology_Seminar_2014-2015
2013-2014: Geometry_and_Topology_Seminar_2013-2014
2012-2013: Geometry_and_Topology_Seminar_2012-2013
2011-2012: Geometry_and_Topology_Seminar_2011-2012
2010: Fall-2010-Geometry-Topology
Dynamics_Seminar_2020-2021