Hamiltonian mean curvature flow

TL;DR

This paper proves that the mean curvature flow of Hamiltonian diffeomorphisms on a compact Riemann surface creates a smooth path in the Hamiltonian diffeomorphism group, supporting a conjecture about Lagrangian submanifolds in Calabi-Yau manifolds.

Contribution

It establishes the smooth evolution of Hamiltonian diffeomorphisms under mean curvature flow on Riemann surfaces, linking to the Thomas-Yau conjecture.

Findings

Mean curvature flow yields smooth paths in Ham(Σ).

Supports the Thomas-Yau conjecture in a specific case.

Provides a new approach to studying Lagrangian submanifolds.

Abstract

Let ({\Sigma}, {\omega}) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on {\Sigma} provides a smooth path in Ham({\Sigma}), the group of all Hamiltonian diffeomorphisms of {\Sigma}. This result gives a proof, in the case of graph of Hamiltonian diffeomorphisms to the conjecture of Thomas and Yau asserting that the mean curvature flow of a compact embedded Lagrangian submanifold S with zero Maslov class in a Calabi- Yau manifolds M exists for all time and converges smoothly to a special Lagrangian submanifold in the Hamiltonian isotopy class of S.