Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in $S^{n-1}$

TL;DR

This paper investigates the uniqueness and regularity of solutions to the fractional harmonic gradient flow on the circle, with values in higher-dimensional spheres, focusing on solutions with small energy levels.

Contribution

It establishes new results on the uniqueness and regularity of solutions in the energy class for fractional harmonic gradient flow on $S^1$, extending previous existence results.

Findings

Proves uniqueness of solutions with small energy

Demonstrates regularity properties of solutions

Extends understanding of fractional harmonic flows in geometric analysis

Abstract

In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.