Topology and Geometry of Legendrian Submanifolds of Sasakian Manifolds

  [NTNU MATH-CAG-MSRC Joint Program on Geometric Analysis] 【10月4日】張樹城 / Topics on Geometry and Analysis of Sasakian Five-Manifolds: Lecture Four

Time  :  15:30-17:00 (Wednesday), 10/4, 2023
Place :    M210, Math. Dept., National Taiwan Normal University

In this lecture series, we will address the related issues on Sasakian geometry including :

     (I) Lecture One : Geometrization Problems in Sasakian 5-Manifolds (9/13)
     (II) Lecture Two : Foliation Minimal Model Program on Sasakian Five-Manifolds (9/20)
     (III) Lecture Three : Topology and Geometry of Legendrian Submanifolds of Sasakian Manifolds (9/27)
     (IV) Lecture Four : Yau Uniformization Conjecture on Complete Noncompact Sasakian Manifolds (10/4)

Lecture Four  Abstract:

    Legendrian submanifolds of contact manifolds and Lagrangian submanifolds of symplectic manifolds are related by symplectization. Furthermore, there is a 1-1 correspondence between minimal Lagrangian cones in a complex Euclidean (n+1)-space and minimal Legendrian submanifolds in (2n+1)-sphere with the canonical contact metric structure. In the SYZ Conjecture, in order to deal with the difficulty which states that most of the special Lagrangian tori fibration have singularities, one can model them locally as special Lagrangian cones in a complex Euclidean 3-space. Such a cone can be characterized by its link of 5-sphere which is a minimal Legendrian surface.

    In this lecture, we will address the related issue such as isotopic Legendrian submanifolds, the Smale conjecture  and existence of minimal Legendrian submanifolds via the Legendrian mean curvature flow.