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

Yifeng Yu

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.

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

Abstract

In this paper, we characterize the geometric structure of the boundary of the minimal level set of the effective Hamiltonian associated with the mechanical Hamiltonian in dimension , where on has a unique maximum and Hessian at this maximizer has two distinct negative eigenvalues. For , the geometry of the level sets of 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 . In particular, we show that does not lie on any flat edge if and only if is differentiable at and its outer normal direction is irrational, except possibly at one exceptional pair of points . Consequently, flat edges are dense along . We also construct an example demonstrating that this exceptional pair can occur, showing the result is sharp.
Paper Structure (18 sections, 31 theorems, 303 equations, 13 figures)

This paper contains 18 sections, 31 theorems, 303 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Global characteristics
  4. More Properties for $H(p,x)={1\over 2}|p|^2+V(x)$
  5. Aubry Set
  6. Mather set
  7. Sketch of the Proof: Comparison between $F_c$ and $\partial F_0$
  8. Homoclinic orbits
  9. Extension and Unit-reparameterization of Orbits
  10. Proof of Theorem \ref{['main']}
  11. Proof of the "Only if" part
  12. Proof of "If" part
  13. Proof of Theorem \ref{['main2']}
  14. Local behavior near the critical point
  15. Examples
  16. ...and 3 more sections

Key Result

Theorem 1.1

Given $n=2$ and Assumption (M), then one of the following holds: (1) For any $p\in \partial F_0$, $p$ is a nonlinear point if and only if$\partial F_0$ is differentiable at $p$ and the associated outer unit normal vector $q_p\in S^1\backslash {\mathbb R}{\mathbb Z}^2$; or (2) There exists a ration

Figures (13)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 8 more figures

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Corollary 2.1
  • Lemma 2.4
  • Remark 2.1
  • ...and 38 more