https://neckpinch.com/authors/glen-wheeler/New arXiv papers by Glen Wheeler, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.18504arxiv:2603.18504Thu, 19 Mar 2026 00:00:00 GMTPhilip Schrader, Glen Wheeler, Valentina Wheeler — The well-known curve shortening flow can be formulated as the gradient flow of the length functional on the space of immersed closed planar curves, where the gradient is taken with respect to a reparametrisation-invariant L² Riemannian metric. This metric is degenerate, giving a geodesic distance of zero between any two curves. We instead consider a family of Sobolev H¹ metrics depending on two parameters λ>0 and a∈ ℝ, where λ sets the weight of the first-derivative term, and a indexes a length normalisation which ensures that the metric is scale-homogeneous. For each such metric, the gradient of length can be written explicitly in terms of a convolution with respect to normalised arc length against the periodic Green's function of (λ² ∂ₓ²−1). The associated evolution is a reparametrisation invariant nonlocal ODE whose right-hand side is well-defined even on curves that are not immersed. Working in the optimal low-regularity setting W^(1,1)(S,ℝ²), we prove local well-posedness using the Picard–Lindelöf theorem and convergence to constant maps in finite time when a<2, and as t→∞ when a≥ 2. This behaviour is exhibited by round circles, which evolve self-similarly and collapse at an explicit time. We further prove that if the initial curve is an immersion, C¹, C², or bounds a strictly convex set, then each of these properties is preserved along the flow.Curve shorteningSingularitiesHomogeneous spaces https://arxiv.org/abs/2507.12088arxiv:2507.12088Wed, 16 Jul 2025 00:00:00 GMTShuhui He, Ben Whale, Glen Wheeler, Valentina-Mira Wheeler — In this paper we propose and study a curvature-based mathematical model for dorsal closure in embryonic drosophila. Using an analysis that mixes maximum-principle and integral-estimates, we establish global existence and convergence for data that mimics the initial geometry of a dorsal closure event. Further, we present a numerical approximation scheme for the flow, establishing stability, consistence, and convergence. We also give sample simulations of the flow with initial configurations that include experimentally observed data.Other curvature flowsStability & uniquenessLong-time existence & convergenceNumerics & visualization https://arxiv.org/abs/2411.18951arxiv:2411.18951Thu, 28 Nov 2024 00:00:00 GMTSamuel Cuthbertson, Glen Wheeler, Valentina Wheeler — 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.Curve shorteningOther curvature flowsLong-time existence & convergence https://arxiv.org/abs/2411.03579arxiv:2411.03579Wed, 06 Nov 2024 00:00:00 GMTSam Cuthbertson, Glen Wheeler, Valentina-Mira Wheeler — In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the L^(∞) sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows along the same lines of Gage and Hamilton, in that we study a rescaling to prove curvature bounds. We additionally show that the influence of an ambient force field may make such a result untrue, by giving sufficient conditions on the ambient field that ensures eventual non-convexity of an initially convex curve evolving under the flow.Curve shortening