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One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds

Jiayin Liu, Shijin Zhang, Yuan Zhou

TL;DR

The paper addresses the one-dimensional asymptotic Plateau problem on $n$-dimensional asymptotically conical manifolds by proving the existence of a geodesic line with prescribed antipodal ideal boundaries, under properness of the exponential map and a fixed opening angle. It develops a high-dimensional min-max framework, constructing an $(n-1)$-parameter sweep-out from length-minimising geodesics on standard cones, and uses a truncated energy gradient flow to obtain a sequence of min-max geodesic segments with uniform proximity to the base point and controlled length. A uniform distance bound and a non-twisting angle estimate enable passage to a limit geodesic line whose asymptotic boundary matches the prescribed antipodal pair, with a Morse index bound of $\le n-1$. This extends the Carlotto–De Lellis result from dimension $2$ to higher dimensions and relaxes curvature assumptions, contributing a robust variational approach to asymptotic minimal geometry in noncompact settings.

Abstract

Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension $n\ge 2$, opening angle $α\in (0,π/2) \setminus \{\arcsin \frac{1}{2k+1}\}_{k \in \mathbb{N}}$ and positive asymptotic rate. Under the assumption that the exponential map is proper at each point, we give a solution to the one dimensional asymptotic Plateau problem on $M$. Precisely, for any pair of antipodal points in the ideal boundary $\partial_\infty M = \mathbb S^{n-1}$, we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index $\le n-1$.

One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds

TL;DR

The paper addresses the one-dimensional asymptotic Plateau problem on -dimensional asymptotically conical manifolds by proving the existence of a geodesic line with prescribed antipodal ideal boundaries, under properness of the exponential map and a fixed opening angle. It develops a high-dimensional min-max framework, constructing an -parameter sweep-out from length-minimising geodesics on standard cones, and uses a truncated energy gradient flow to obtain a sequence of min-max geodesic segments with uniform proximity to the base point and controlled length. A uniform distance bound and a non-twisting angle estimate enable passage to a limit geodesic line whose asymptotic boundary matches the prescribed antipodal pair, with a Morse index bound of . This extends the Carlotto–De Lellis result from dimension to higher dimensions and relaxes curvature assumptions, contributing a robust variational approach to asymptotic minimal geometry in noncompact settings.

Abstract

Let be an asymptotically conical Riemannian manifold having dimension , opening angle and positive asymptotic rate. Under the assumption that the exponential map is proper at each point, we give a solution to the one dimensional asymptotic Plateau problem on . Precisely, for any pair of antipodal points in the ideal boundary , we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index .

Paper Structure

This paper contains 17 sections, 25 theorems, 359 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Ideas of the proof of of Theorem \ref{['ndimanti']}
  3. Notations
  4. Geometry of standard cones
  5. Geometry of $\mathcal{C}_\alpha^2$
  6. Geometry of $\mathcal{C}_\alpha^n$ with $n \ge 3$
  7. Asymptotically conical manifolds
  8. A key convergence lemma under properness of exponential map
  9. Existence of min-max geodesic segments
  10. Construction of homotopy class
  11. Min-max value and lower/upper bound
  12. Truncated energy functional and its gradient flow
  13. A mountain pass theorem for geodesic segments
  14. Existence of min-max geodesic lines
  15. Uniform estimate for distances between $o$ and minmax geodesic segments
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be an asymptotically conical surface with nonnegative Gaussian curvature. Then, for any pair of antipodal points $a,-a \in \partial_\infty M = \mathbb S^{1}$, there exists an embedded geodesic line $\Gamma$ with asymptotic prescribed boundary $\partial_\infty \Gamma = \{a,-a\}$ and Morse

Figures (11)

  • Figure 1: $M$ has nontrivial topology or $M$ has boundary.
  • Figure 2: Geodesic segments joining $p_j$ and $q_j$.
  • Figure 3: An illustration of $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$.
  • Figure 4: An illustration of $\Gamma \in \Gamma_{p,q}$.
  • Figure 5: $\angle(-\dot{\bar{{\gamma}}}(1),\bar{b})$ and $\angle(\bar{a},\bar{b})$.
  • ...and 6 more figures

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 51 more