https://neckpinch.com/authors/adam-martens/New arXiv papers by Adam Martens, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.28976arxiv:2603.28976Mon, 30 Mar 2026 00:00:00 GMTAlbert Chau, Adam Martens — In the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that g is only equivalent to a complete bounded curvature metric h while satisfying a Morrey-type condition on the gradient of g relative to h: a local integral condition on the covariant derivative ∇ₕ g. The Morrey-type condition was first considered in in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for g to have unbounded curvature on M. As in, our long-time solution enjoys curvature decay estimates implying in particular that M is diffeomorphic to ℝⁿ.Ricci flowBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2408.11115arxiv:2408.11115Tue, 20 Aug 2024 00:00:00 GMTAdam Martens — In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.Ricci flowSingularitiesScalar curvatureLimit spaces & RCDStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2407.04786arxiv:2407.04786Fri, 05 Jul 2024 00:00:00 GMTAdam Martens — We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than 1/t curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.Ricci flow https://arxiv.org/abs/2403.02564arxiv:2403.02564Tue, 05 Mar 2024 00:00:00 GMTAlbert Chau, Adam Martens — We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical L^p-integrable, this flow converges locally smoothly to a limiting metric g(∞) on M with (M,g(∞)) isometric to the standard flat ℝⁿ, which implies topological rigidity of M. This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.Ricci flowStability & uniquenessBoundaries & non-compactLong-time existence & convergence https://arxiv.org/abs/2307.08088arxiv:2307.08088Sun, 16 Jul 2023 00:00:00 GMTAlbert Chau, Adam Martens — Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow g(t) emerging from an arbitrary 3D complete noncompact Riemannian manifold (M³, g₀) which has nonnegative Ricci curvature. We show g(t) is complete for positive times provided g₀ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show g(t) is complete for positive times provided g₀ is a compactly supported perturbation of a nonnegative sectional curvature metric on ℝ³.Ricci flowOther curvature flows3-manifoldsWeak & singular flowsBoundaries & non-compact