Geometry and Topology Seminar 2019-2020
|September 9||Bing Wang (UW Madison)||"The extension problem of the mean curvature flow"||(Local)|
|September 16||Ben Weinkove (Northwestern University)||"Gauduchon metrics with prescribed volume form"||Lu Wang|
|September 23||Jiyuan Han (UW Madison)||"Deformation theory of scalar-flat ALE Kahler surfaces"||(Local)|
|October 7||Yu Li (UW Madison)||"Ricci flow on asymptotically Euclidean manifolds"||(Local)|
|October 14||Sean Howe (University of Chicago)||"TBA"||Melanie Matchett Wood|
|October 21||Nan Li (CUNY)||"TBA"||Lu Wang|
|October 28||Ronan Conlon||"TBA"||Bing Wang|
|November 4||Jonathan Zhu (Harvard University)||"TBA"||Lu Wang|
|November 7||Gaven Martin (University of New Zealand)||"TBA"||Simon Marshall|
|November 18||Caglar Uyanik (Illinois)||"TBA"||Kent|
|December 2||Peyman Morteza (UW Madison)||"TBA"||(Local)|
Deformation theory of scalar-flat ALE Kahler surfaces
We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kahler surfaces. This is used to prove that for any scalar-flat Kahler ALE surfaces, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of \C^2/\Gamma, where \Gamma is a finite subgroup of U(2) without complex reflections. This is a joint work with Jeff Viaclovsky.
In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem. A classification of diffeomorphism types is also given for all AE 3-manifolds with nonnegative scalar curvature.
The extension problem of the mean curvature flow
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3. A key ingredient of the proof is to show a two-sided pseudo-locality property of the mean curvature flow, whenever the mean curvature is bounded. This is a joint work with Haozhao Li.
Gauduchon metrics with prescribed volume form
Every compact complex manifold admits a Gauduchon metric in each conformal class of Hermitian metrics. In 1984 Gauduchon conjectured that one can prescribe the volume form of such a metric. I will discuss the proof of this conjecture, which amounts to solving a nonlinear Monge-Ampere type equation. This is a joint work with Gabor Szekelyhidi and Valentino Tosatti.
Archive of past Geometry seminars