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)||"TBA"||(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.
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