Lagrangian angles of family of Lagrangian fibrations under mean curvature flow

TL;DR

This paper investigates how Lagrangian angles evolve under mean curvature flow, showing that in the case n=1, the angle function satisfies a degenerated PDE and corresponds to smooth curve foliations.

Contribution

It introduces a PDE characterization of Lagrangian angles under mean curvature flow and links solutions to smooth foliations, providing new insights into their geometric behavior.

Findings

In the case n=1, the angle function satisfies a degenerated PDE.

Smooth solutions correspond to smooth foliations of curves.

The study enhances understanding of Lagrangian fibrations under mean curvature flow.

Abstract

In this paper, we discuss the Lagrangian angles of a family of Lagrangian fibrations moved under mean curvature flow. In the case $n=1$, the angle function is shown to satisfy a degenerated partial differential equation. We prove that any smooth solution to the equation also corresponds to smooth foliation of curves under mean curvature flow.