Geometry of the Minimal Level Set of the Effective Hamiltonian in Two Dimensions

TL;DR

This paper characterizes the geometric structure of the minimal level set boundary of the effective Hamiltonian in two dimensions, revealing conditions for differentiability, the density of flat edges, and the existence of an exceptional pair of points.

Contribution

It provides an explicit, verifiable characterization of the boundary of the minimal level set of the effective Hamiltonian in 2D, filling a gap in the geometric understanding.

Findings

Flat edges are dense along the boundary of the minimal level set.

Differentiability at a point implies an irrational outer normal direction, except at one exceptional pair.

An example shows the exceptional pair can indeed occur, confirming the sharpness of the result.

Abstract

In this paper, we characterize the geometric structure of the boundary of the minimal level set $F_0$ of the effective Hamiltonian $\overline{H}$ associated with the mechanical Hamiltonian \[ H(p,x)=\frac12|p|^2+V(x) \] in dimension $n=2$, where $V$ on $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ has a unique maximum and Hessian at this maximizer has two distinct negative eigenvalues. For $n=2$, the geometry of the level sets of $\overline{H}$ strictly above the minimum has been largely understood since the 1990s, mainly through the equivalent formulation in terms of stable norms; we fill the remaining gap at the minimal level by providing an explicit, verifiable characterization of $\partial F_0$. In particular, we show that $p \in \partial F_0$ does not lie on any flat edge if and only if $\partial F_0$ is differentiable at $p$ and its outer normal direction is irrational, except possibly at one exceptional pair of points $\pm p_0$. Consequently, flat edges are dense along $\partial F_0$. We also construct an example demonstrating that this exceptional pair can occur, showing the result is sharp.