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A Ginzburg-Landau problem on a circular cone

Christian Cofoid, Dmitry Golovaty, Etienne Sandier, Peter Sternberg

TL;DR

This work analyzes minimizers of a Ginzburg-Landau energy for tangent vector fields on a circular cone, focusing on how a boundary degree $\bar{d}$ and the cone opening angle $\alpha$ govern vortex formation. A cone-adapted vortex-ball framework yields a sharp energy expansion with a leading $\pi m(\bar{d},\alpha)\log(1/\varepsilon)$ term and a renormalized energy $W(\mathbf{a};\mathbf{d},\alpha)$ that selects vortex locations, including a mandatory vortex at the tip. The minimizers converge to a canonical harmonic unit vector field away from the vortices, and the degrees are chosen to minimize the effective energy $m(\bar{d},\alpha)$. Overall, the paper extends BBH-type analysis to singular surfaces, revealing fractional degrees and tip-defect behavior that depend on $\alpha$, with implications for nematic textures on conical geometries.

Abstract

We carry out an asymptotic analysis for a Ginzburg-Landau type model for tangent vector fields defined on a cone. The results, in the spirit of Brezis, Bethuel and Helein, establish the degree and asymptotic location of vortices, one of which must be situated at the tip of the cone.

A Ginzburg-Landau problem on a circular cone

TL;DR

This work analyzes minimizers of a Ginzburg-Landau energy for tangent vector fields on a circular cone, focusing on how a boundary degree and the cone opening angle govern vortex formation. A cone-adapted vortex-ball framework yields a sharp energy expansion with a leading term and a renormalized energy that selects vortex locations, including a mandatory vortex at the tip. The minimizers converge to a canonical harmonic unit vector field away from the vortices, and the degrees are chosen to minimize the effective energy . Overall, the paper extends BBH-type analysis to singular surfaces, revealing fractional degrees and tip-defect behavior that depend on , with implications for nematic textures on conical geometries.

Abstract

We carry out an asymptotic analysis for a Ginzburg-Landau type model for tangent vector fields defined on a cone. The results, in the spirit of Brezis, Bethuel and Helein, establish the degree and asymptotic location of vortices, one of which must be situated at the tip of the cone.

Paper Structure

This paper contains 7 sections, 13 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower Bound for $E_\varepsilon(\cdot, \mathcal{C}_\alpha)$
  4. Upper Bound for $\min_u E_\varepsilon(u,\mathcal{C}_\alpha)$
  5. Definition of the sequence yielding the asymptotic upper bound
  6. Upper bound: Calculation of the energy of the constructed sequence
  7. Limiting harmonic map and renormalized energy

Key Result

Lemma 3.1

If $u\in H^1_g$, then $|u|$ is not bounded away from zero on any punctured ball of $\mathcal{C}_\alpha$ centered at the origin.

Figures (3)

  • Figure 3.1: The merging of two balls, grown respectively from $B_0\in\mathcal{B}_0$ and $B_1\in\mathcal{B}_0$ to obtain $B\in \mathcal{B}(t_0)$ of radius $r(B) = e^{t_0}\left(r(B_0) + r(B_1)\right)$. To reduce clutter in this and subsequent figures, we drop the supercript indicating time in the notation for balls.
  • Figure 3.2: The merging of two balls, grown respectively from the central ball $B_0\in\mathcal{B}_0$ and $B_1\in\mathcal{B}_0$ to obtain the new central ball $B_0\in \mathcal{B}(t_0)$ of radius $e^{t_0}\left(r(B_0) + 2r(B_1)\right).$
  • Figure 3.3: Growth of three balls $B_0,B_1,B_2\in\mathcal{B}_0$ generates three new balls with the larger radii and the same respective centers. The balls grown from $B_1$ and $B_2$ touch at the time $t_0$ and have to be replaced by a larger ball $B_s$ with the radius $e^{t_0}\left(r(B_1) + r(B_2)\right).$ Because $B_s$ intersects the central ball grown from $B_0$, these balls are merged in a larger new central ball of radius $e^{t_0}\left(r(B_0) + 2r(B_1) + 2r(B_2)\right).$

Theorems & Definitions (28)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Definition 3.5
  • Lemma 3.6
  • proof
  • ...and 18 more