Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generic density of stationary geodesic nets that are not closed geodesics

Talant Talipov

TL;DR

The paper addresses the Nabutovsky-Parsch conjecture in dimensions $n\ge 3$ by proving that for a $C^ty$-generic metric on a closed manifold, the union of essential embedded stationary geodesic nets is dense in the manifold. The approach combines the Liokumovich–Staffa density result with localized, $C^k$-small conformal perturbations that produce essential nets meeting any prescribed open set, using two geometric constructions (eyeglass nets and twisted figure-eight nets) depending on how two close closed geodesics interact. Openness of the relevant metric-nerve property follows from the Structure Theorem and the Inverse Function Theorem, while genericity is obtained via the Bumpy Metrics Theorem to attain nondegeneracy. The result strengthens the Nabutovsky–Parsch conjecture by ensuring ubiquitous presence and distribution of nontrivial stationary geodesic networks under generic metrics, with perturbations that are localized and controllable.

Abstract

We prove that for a Baire-generic Riemannian metric on a closed smooth manifold of dimension greater than or equal 3, the union of stationary geodesic nets that are not closed geodesics forms a dense set. This result confirms a Nabutovsky-Parsch conjecture in this case.

Generic density of stationary geodesic nets that are not closed geodesics

TL;DR

The paper addresses the Nabutovsky-Parsch conjecture in dimensions by proving that for a -generic metric on a closed manifold, the union of essential embedded stationary geodesic nets is dense in the manifold. The approach combines the Liokumovich–Staffa density result with localized, -small conformal perturbations that produce essential nets meeting any prescribed open set, using two geometric constructions (eyeglass nets and twisted figure-eight nets) depending on how two close closed geodesics interact. Openness of the relevant metric-nerve property follows from the Structure Theorem and the Inverse Function Theorem, while genericity is obtained via the Bumpy Metrics Theorem to attain nondegeneracy. The result strengthens the Nabutovsky–Parsch conjecture by ensuring ubiquitous presence and distribution of nontrivial stationary geodesic networks under generic metrics, with perturbations that are localized and controllable.

Abstract

We prove that for a Baire-generic Riemannian metric on a closed smooth manifold of dimension greater than or equal 3, the union of stationary geodesic nets that are not closed geodesics forms a dense set. This result confirms a Nabutovsky-Parsch conjecture in this case.

Paper Structure

This paper contains 3 sections, 1 theorem, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geodesic nets
  3. Proof of Main Theorem

Key Result

Proposition 3.1

Suppose $n \geq 3$ and let $k \in \mathbb{Z}_{> 0}$ and $U \subset M$ be a nonempty open set. The set $\mathcal{M}^k_U$ of $C^k$-smooth Riemannian metrics on $M^n$ for which there exists a nondegenerate, essential, embedded stationary geodesic net $\gamma$ intersecting $U$ is open and dense in the $

Figures (6)

  • Figure 1: Stationary twisted figure-eight (left) and stationary eyeglass (right)
  • Figure 2: Geodesics $\alpha$ and $\rho$ near $a$
  • Figure 3: Triple junction at $a_t$
  • Figure 4: Geodesics $\alpha$ and $\beta$ near $v$
  • Figure 5: Perturbation of $\alpha$
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop']}
  • proof : Proof of Main Theorem