[Paper Review] On the Regularity of Hamiltonian Stationary Lagrangian manifolds

Jingyi Chen, Micah Warren|arXiv (Cornell University)|Nov 8, 2016

Geometric Analysis and Curvature Flows15 references4 citations

TL;DR

This paper establishes that any $C^1$ Hamiltonian stationary Lagrangian submanifold in $\mathbb{C}^n$ is necessarily real analytic, proving a fourth-order analogue of Morrey’s theorem for minimal submanifolds. The authors show that weak solutions to the fourth-order Hamiltonian stationary equation with $C^{1,1}$ potential functions and bounded Hessian are smooth, under convexity and smallness conditions, via interior estimates and coordinate rotation techniques.

ABSTRACT

We prove a Morrey-type theorem for Hamiltonian stationary submanifolds of $\mathbb{C}^{n}$. Namely, if $L$ $\subset$ $\mathbb{C}^{n}$ is a $C^{1}$ Lagrangian submanifold with weakly harmonic Lagrangian phase $θ,$ then $L$ must be smooth. In the process we also discuss a local version of the equation, which is a nonlinear fourth order double divergence equation of the potential function whose gradient graph defines the Hamiltonian stationary submanifolds locally, and we establish full regularity and removability of singular sets of capacity zero for weak solutions with $C^{1,1}$ norm below a dimensional constant.

Motivation & Objective

• To establish the regularity of Hamiltonian stationary Lagrangian submanifolds in $\mathbb{C}^n$ when the potential function is only $C^1$.
• To extend Morrey’s classical result on minimal submanifolds to the fourth-order setting of Hamiltonian stationary Lagrangians.
• To prove that weak solutions to the Hamiltonian stationary equation with $C^{1,1}$ potential functions are smooth under convexity and smallness conditions on the Hessian.
• To demonstrate removability of singular sets of capacity zero for weak solutions to the fourth-order equation.
• To show that the Lagrangian phase function being weakly harmonic implies full regularity under appropriate geometric constraints.

Proposed method

• Use the variational formulation of Hamiltonian stationary Lagrangian submanifolds via the volume functional under Hamiltonian variations.
• Characterize Hamiltonian stationary submanifolds locally as gradient graphs of functions $u$ satisfying a fourth-order nonlinear elliptic equation derived from the Euler-Lagrange equation.
• Employ the geometric formulation where the Lagrangian phase function $\theta = \operatorname{Im} \log \det(I + iD^2u)$ satisfies $\Delta_g \theta = 0$, a second-order equation on the graph.
• Apply coordinate rotation techniques to transform the Hessian into a uniformly convex form, enabling application of second-order elliptic theory.
• Use bi-Lipschitz coordinate changes to transfer interior estimates from rotated to original coordinates, preserving regularity.
• Leverage the fact that $\theta$ is odd in $D^2u$ to reduce the problem to analyzing $-u$, which becomes uniformly convex under rotation, allowing application of known regularity results.

Experimental results

Research questions

• RQ1 Can the Morrey-type regularity result for minimal submanifolds be extended to the fourth-order setting of Hamiltonian stationary Lagrangian submanifolds?
• RQ2 Under what conditions on the Hessian of the potential function $u$ does a weak solution to the Hamiltonian stationary equation become smooth?
• RQ3 Can singular sets of capacity zero be removed for weak solutions to the fourth-order Hamiltonian stationary equation?
• RQ4 How does the regularity of the Lagrangian phase function $\theta$ relate to the regularity of the potential function $u$?
• RQ5 What geometric and analytic conditions ensure that a $C^1$ Lagrangian submanifold critical under Hamiltonian variations is real analytic?

Key findings

• Any $C^1$ Hamiltonian stationary Lagrangian submanifold in $\mathbb{C}^n$ is real analytic, establishing a fourth-order analogue of Morrey’s theorem.
• Weak solutions to the Hamiltonian stationary equation with $C^{1,1}$ potential functions and $\|u\|_{C^{1,1}} \leq c(n)$ are smooth, provided the Hessian satisfies a convexity condition.
• Interior $C^{k,\alpha}$ estimates for all $k$ can be obtained for the potential function $u$ when the Hessian is bounded and the phase function is weakly harmonic.
• The singular set of a weak solution to the Hamiltonian stationary equation has capacity zero and is removable, meaning the solution extends smoothly across it.
• After a suitable rotation of coordinates, the potential function becomes uniformly convex, enabling the use of second-order elliptic theory to derive full regularity.
• The result holds even when $D^2u$ is only continuous or has mild discontinuities, as long as $\|u\|_{C^{1,1}} \leq c(n)$ with $c(n)$ small enough depending on dimension.