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On the first width of hyperbolic surfaces: multiplicity and lower bounds

Vanderson Lima

TL;DR

The work analyzes the first width $\omega_1$ for closed negatively curved surfaces via Allen-Cahn and Almgren-Pitts min-max theories, proving that limiting configurations are multiplicity-one geodesic networks consisting of a primitive figure-eight plus disjoint simple closed geodesics. It establishes a sharp universal lower bound for $\omega_1$ on hyperbolic surfaces and shows this bound is asymptotically attained in families with small systoles, while also computing $\omega_1$ for the Bolza surface. The authors derive explicit pants-sweepouts, demonstrate non-connected limit interfaces, and connect the phase-transition width with classical width, providing concrete geometric realizations and asymptotics. These results illuminate the structure of the first volume-width analogue in two dimensions and reveal precise geometric configurations realizing minimal first widths on hyperbolic surfaces.

Abstract

On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs with multiplicity one. As an application we obtain a uniform sharp lower bound for the first min-max width of closed hyperbolic surfaces and prove it is only attained asymptotically. Moreover, we compute the first width of the Bolza surface and of some hyperbolic surfaces with small systoles.

On the first width of hyperbolic surfaces: multiplicity and lower bounds

TL;DR

The work analyzes the first width for closed negatively curved surfaces via Allen-Cahn and Almgren-Pitts min-max theories, proving that limiting configurations are multiplicity-one geodesic networks consisting of a primitive figure-eight plus disjoint simple closed geodesics. It establishes a sharp universal lower bound for on hyperbolic surfaces and shows this bound is asymptotically attained in families with small systoles, while also computing for the Bolza surface. The authors derive explicit pants-sweepouts, demonstrate non-connected limit interfaces, and connect the phase-transition width with classical width, providing concrete geometric realizations and asymptotics. These results illuminate the structure of the first volume-width analogue in two dimensions and reveal precise geometric configurations realizing minimal first widths on hyperbolic surfaces.

Abstract

On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs with multiplicity one. As an application we obtain a uniform sharp lower bound for the first min-max width of closed hyperbolic surfaces and prove it is only attained asymptotically. Moreover, we compute the first width of the Bolza surface and of some hyperbolic surfaces with small systoles.

Paper Structure

This paper contains 14 sections, 17 theorems, 114 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Min-max theory
  3. The $k$-widths
  4. Almgren--Pitts theory
  5. Phase transition theory
  6. Comparison between the min-max theories
  7. The Multiplicity one result
  8. Main results
  9. An auxiliar Lemma
  10. Applications for hyperbolic surfaces
  11. A sweepout of a pair of pants
  12. Sharpness of the width lower bound
  13. Existence non-connected limit interfaces
  14. The width of the Bolza surface

Key Result

Theorem 1

Let $(M,g)$ be a closed oriented Riemannian surface of genus $m \geq 2$, such that $K_{g} < 0$. Then, there exist a primitive figure eight geodesic $\sigma_{\infty}$ and a possibly empty collection of primitive simple closed geodesics $\sigma_{1},\dots,\sigma_{N}$, $N \leq 3m-3$, which are disjoint Moreover,

Figures (8)

  • Figure 1: $\Sigma\cap(B_{\rho}(p_{\ast})\backslash \overline B_{2r}(p_{\ast}))$.
  • Figure 2: $\Sigma\backslash \overline B_{\rho/2}(p_{\ast})$.
  • Figure 3: A figure eight geodesic in a pair of pants.
  • Figure 4: Step 2 in the construction of the sweepout $t \to [[\sigma_t]]$.
  • Figure 5: The pair of pants $\mathrm{Y}_a$.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition
  • Definition
  • Lemma 1: Li Li, Chodosh-Mantoulidis CM2
  • Definition
  • Proposition 1
  • Definition
  • ...and 16 more