https://neckpinch.com/authors/jonathan-j-zhu/New arXiv papers by Jonathan J. Zhu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.19648arxiv:2609.19648Thu, 17 Sep 2026 00:00:00 GMTKahnrad Braxton, Tang-Kai Lee, Jonathan J. Zhu — We prove the linear stability of an embedded mean curvature flow shrinker in ℝ⁴ with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.Mean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://arxiv.org/abs/2310.08005arxiv:2310.08005Thu, 12 Oct 2023 00:00:00 GMTSven Hirsch, Jonathan J. Zhu — We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding-Minicozzi respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds.Mean curvature flowVolume-preserving & constrained flowsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://arxiv.org/abs/2310.02649arxiv:2310.02649Wed, 04 Oct 2023 00:00:00 GMTPaul Bryan, Mat Langford, Jonathan J. Zhu — We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof that simple spherical curves either contract to round points (in finite time) or converge to great circles (in infinite time).Curve shorteningSingularities https://arxiv.org/abs/2302.14258arxiv:2302.14258Tue, 28 Feb 2023 00:00:00 GMTMat Langford, Jonathan J. Zhu — We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to prove that any such flow either converges in infinite time to a (unique) "critical chord", or contracts in finite time to a "round half-point" on the boundary.Curve shorteningSingularitiesBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2108.02326arxiv:2108.02326Thu, 05 Aug 2021 00:00:00 GMTAo Sun, Jonathan J. Zhu — We show that S²× S² is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for S²× N, where N belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.Ricci flowMean curvature flowRicci solitonsEinstein metricsSmale conjecture & diffeomorphismsStability & uniqueness