The talk will begin with a brief review on the Barden-Smale classification of simply connected 5-manifolds. This will follow by results on topological characterization of a simply connected (positive) Sasakian 5-manifold. Toward the end, I will discuss some recent progress in topology of moduli of positive Sasakian structures.
The first part of my talk is an introduction to the isoperimetric inequality, following Osserman's survey. Next I would present the joint work with Bo-Hshiung Wang in which we confirm two two special cases for the convex body isoperimetric conjecture in the plane.
The base of a Sasakian 5-manifold M is a complex V-manifold S. By using the language of Seifert bundle, one obtains a log del Pezzo surface (S,D) from M, where D is a suitably chosen divisor. In this talk, we introduce and explain the algebraic geometric language that are essential for a classification of such a pair (S,D). This includes the theory of divisors, intersection numbers, surface quotient singularities (=klt=Kawamata log terminal singularities), and surface minimal model program.
We propose a method of classification of Sasakian 5-manifolds according to the global properties of the Reeb foliation. The transverse Kaehler structure of a quasi-regular Sasakian manifold pushes down to a Kaehler structure on a compact orbifold surface equipped with a Kaehler metric. Then, by the geometric nature of classification problems, in particular in view of Hamilton-Perelman-Yau program for the Ricci flow, Song-Tian’s analytic minimal model program for the Kaehler-Ricci flow, we focus on the method of Sasaki-Ricci flow to attach it along this direction.
The aim of this talk is to study the existence geometric problem for canonical Sasakian metrics including
1. Yau-Tian-Donaldson Conjecture on transverse Fano Sasakian manifolds
2. Yau's Uniformization Conjecture on complete noncompact Sasakian manifolds
3. The Legendrian mean curvature flow on Sasakian manifolds