Foliation Minimal Model Program on Sasakian Five-Manifolds

  [NTNU MATH-CAG-MSRC Joint Program on Geometric Analysis] 【09月20日】張樹城 / Topics on Geometry and Analysis of Sasakian Five-Manifolds: Lecture Two

Time  :  15:30-17:00 (Wednesday), 09/20, 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  Two  Abstract:

    In 1982, R. Hamilton introduced the Ricci flow and then by studying the singularity models of Ricci flow, G. Perelman completely solved Thurston geometrization conjecture and Poincare conjecture for a closed 3-manifold in 2002 and 2003. On the other hand, Mori minimal model program in birational geometry can be viewed as the complex analogue of Thurston's geometrization conjecture. In 1985, H.-D. Cao introduced the Kaehler-Ricci flow and then recaptured the Calabi-Yau Conjecture. Recently, there is a conjecture picture by Song-Tian that the Kaehler-Ricci flow should carry out the minimal model program with scaling on projective varieties. Song-Weinkove established the above conjecture on a projective algebraic surface.

     The Sasaki-Ricci flow is introduced by Smoczyk-Wang-Zhang to study the existence of Sasaki-Einstein metrics on Sasakian manifolds. It can be viewed as a Sasaki analogue of Cao's result for the Kaehler-Ricci flow. It is natural to conjecture that the Sasaki-Ricci flow will carry out the foliation minimal model program with scaling on quasi-regular Sasakian 5-manifolds as well. Indeed, Chang-Lin-Wu proved the Sasaki analogue of minimal model program on closed quasi-regular Sasakian 5-manifolds of foliation cyclic quotient singularities.