Workshop

Various geometric evolution equations and function theory have the common goal of understanding the relations between the geometry, analysis, and topology of manifolds, submanifolds, vector bundles, maps, and other geometric structures. The fields of geometry, analysis, and topology are synthesized through the study of geometric and topological invariants via a priori estimates. The goal of the Introductory Workshop is to survey current and recentdevelopments in geometric evolution equations and function theory in real and complex geometry. Invited speakers include: Albert Chau (University of Waterloo), Jaigyoung Choe (Seoul National University), Andre Neves (Princeton University), Lei Ni (University of California, San Diego), Natasa Sesum (New York University, Courant Institute), Jian Song (Johns Hopkins University), Jiaping Wang (University of Minnesota), Ben Weinkove (Imperial College, London), and Zhou Zhang (Massachusetts Institute of Technology). The talks will focus on the following topics: Geometric flows in complex geometry Function theory on manifolds Ricci flow and canonical metrics Mean curvature flow and applications Schedule of the Lectures Monday September 11 9:30-10:30am Ben Weinkove: Geometric flows and moment maps, part I Tea Break 11am-12pm Jiaping Wang: Stability of harmonic functions  Abstract: We aim to explain a result of P. Li and L. Tam relating the harmonic functions to the number of ends of a complete manifold. We start with their constructive proof of the existence of Green's function, and use the result to show the stability of the space of harmonic functions under the perturbation within a compact domain. We then reach the conclusion by carrying out a straightforward construction of the so-called barrier function on each end. Lunch 2-3pm Andre Neves: Lagrangian mean curvature flow, part I Tea Break 3:30-4:30pm Natasa Sesum: Convergence and stability results for the Ricci flow, part I Tuesday September 12 9:30-10:30am Ben Weinkove: Geometric flows and moment maps, part II Tea Break 11am-12pm Jiaping Wang: Sharp estimate of Green's function and applications  Abstract: We will establish a sharp integral decay estimate of the minimal positive Green's function on a complete manifold with positive spectrum. Some applications will also be discussed. Lunch 2-3pm Natasa Sesum: Convergence and stability results for the Ricci flow, part II Tea Break 3:30-4:30pm Lei Ni: Ricci flow and invariant cones Wednesday September 13 9:30-10:30am Jaigyoung Choe: Rigidity of Capillary Surfaces and First Eigenvalue of Minimal Surfaces Tea Break 11am-12pm Andre Neves: Lagrangian mean curvature flow, part II Lunch No talks in the afternoon Thursday September 14 9:30-10:30am TBA Tea Break 11am-12pm Jian Song: The Kaehler-Ricci flow on surfaces, part I  Abstract: We study the Kaehler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical metric is a generalized Kaehler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for Kaeher surfaces with a numerically effective canonical line bundle by the Kaehler-Ricci flow. In general, we propose to find canonical metrics on the canonical models of projective varieties of positive Kodaira dimension. This is a joint work with G. Tian. Lunch 2-3pm Albert Chau: Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part I Friday September 15 9:30-10:30am Jian Song: The Kaehler-Ricci flow on surfaces, part II Tea Break 11am-12pm Albert Chau: Uniformization of complete non-compact Kaehler manifolds and the Kaehler Ricci flow, part II Lunch 2-3pm Zhou Zhang: Kaehler-Ricci Flows over Manifolds of General Type