Geometric analysis of Lagrangian mean curvature flows and Ricci flows

2017 Fiscal Year Final Research Report

• Project/Area Number: 16H07229
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Yamamoto Hikaru 東京理科大学, 理学部第一部数学科, 助教 (50778173)
• Project Period (FY): 2016-08-26 – 2018-03-31
• Keywords: mean curvature flow / Lagrangian / special Lagrangian / Ricci flow / mirror symmetry / dHYM connection

Outline of Final Research Achievements

I proposed a method to construct a special Lagrange submanifold in the lattice quotient of the tangential bundle of a tropical manifold. I also proved that the Fourier-Mukai transform of this special Lagrange submanifold is a deformed Hermitian Yang Mills connection with support on a complex submanifold in the mirror. A paper on these results will be published in Math. Z.
I proved that if the second fundamental form of a self-shrinker satisfying the pinching condition takes zero at some point, then it becomes a plane. As an application, if a codimension 1 mean curvature flow with initial pinching condition in Euclidean space develops finite time singularities, then all general type I singularities are actually special type I singularities. The proof is summarized in the RIMS Kokyuroku.

Free Research Field

Differential Geometry