A curvature flow that deforms curves to an embedded target

TL;DR

This paper introduces a new curvature flow called the target flow, which deforms arbitrary embedded curves to a specified target curve, providing a solution to a problem posed by Yau.

Contribution

The paper develops the target flow with ambient forcing and proves its smooth convergence to the target curve for a broad class of initial data.

Findings

The target flow successfully deforms source curves to target curves.

Convergence to the target is smooth under certain conditions.

Broadens the class of source and target curves for Yau's problem.

Abstract

In this paper we introduce the target flow -- a specific curve shortening flow with an ambient forcing term -- that, given an embedded (not necessarily convex) target curve, will attempt to evolve a given source curve to that target. The motivation for this flow is to address a question of Yau. Our main result is that the target flow with uniformly normal graphical data converges smoothly to the target, broadening the class of known sources and targets such that Yau's problem has a solution.