An energy gap phenomenon for the Whitney sphere

TL;DR

This paper proves a gap theorem for the Whitney sphere as a solution to a specific differential equation involving the Lagrangian trace-free second fundamental form in complex Euclidean space.

Contribution

It establishes a new gap theorem characterizing the Whitney sphere among solutions to a certain geometric PDE for Lagrangian surfaces.

Findings

The Whitney sphere satisfies the equation $ abla^*T=0$.

A gap phenomenon is demonstrated for solutions close to the Whitney sphere.

The result characterizes the Whitney sphere uniquely under the given PDE.

Abstract

For an immersed Lagrangian submanifold, let $\check{A}$ be the Lagrangian trace-free second fundamental form. In this note we consider the equation $\nabla^*T=0$ on Lagrangian surfaces immersed in $\mathbb{C}^2$, where $T=-2\nabla^*(\check{A}\lrcorner\omega)$, and we prove a gap theorem for the Whitney sphere as a solution to this equation.