https://neckpinch.com/authors/jim-isenberg/New arXiv papers by Jim Isenberg, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2502.08500arxiv:2502.08500Wed, 12 Feb 2025 00:00:00 GMTJames Isenberg, Dan Knopf, Zilu Ma, Natasa Sesum — We demonstrate that any four-dimensional shrinking Ricci soliton (𝓑 × S², g), where 𝓑 is any two-dimensional complete noncompact surface and g is a warped product metric over the base 𝓑, has to be isometric to the generalized cylinder ℝ²×S² equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products – but not products – and provide rigorous examples of the formation of generalized cylinder singularity models ℝ^k×S^ℓ.Ricci flowRicci solitonsSingularities4-manifoldsSurfacesBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2403.06427arxiv:2403.06427Mon, 11 Mar 2024 00:00:00 GMTDavid Garfinkle, James Isenberg, Dan Knopf, Haotian Wu — Kröncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Krö20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].Ricci flowSingularitiesStability & uniquenessNumerics & visualization https://arxiv.org/abs/2303.04800arxiv:2303.04800Wed, 08 Mar 2023 00:00:00 GMTEric Bahuaud, Christine Guenther, James Isenberg, Rafe Mazzeo — We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on our recent work where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric.Ricci flowSingularitiesStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trickRotational symmetry & warped products https://arxiv.org/abs/2206.01890arxiv:2206.01890Sat, 04 Jun 2022 00:00:00 GMTDavid Garfinkle, James Isenberg, Dan Knopf, Haotian Wu — In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that for small rotationally-symmetric perturbations of the initial embeddings that are sufficiently far away from the tip, MCF develops Type-I "neckpinch" singularities. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally-symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence – becoming round – and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this work is an adaptation of the "overlap" method introduced in [GIKW21] and permits angular dependence.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsNumerics & visualization