https://neckpinch.com/authors/dain-kim/New arXiv papers by Dain Kim, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2611.00139/arxiv:2611.00139Thu, 29 Oct 2026 00:00:00 GMTDain Kim, Tristan Ozuch — We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection ±1. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is C¹⁴-open. It is nonempty if and only if the manifold contains a sphere of self-intersection ±1. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct C¹⁴-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard kℂP}²\#ℓℂP}², through finitely many blowdown singularities. (2) On the standard 2ℂP}²\#9ℂP}², there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.Ricci flowRicci solitonsSingularities4-manifoldsBoundaries & non-compactEigenvalues & spectrum https://neckpinch.com/papers/2510.21997/arxiv:2510.21997Fri, 24 Oct 2025 00:00:00 GMTDain Kim, Tristan Ozuch — We prove that on ALF n-manifolds with n≥ 4 the Ricci flow preserves the ALF structure, and develop a weighted Fredholm framework adapted to ALF manifolds. Motivated by Perelman's λ-functional, we define a renormalized functional λ_ALF whose gradient flow is the Ricci flow. It is built from a relative mass with respect to a reference Ricci-flat metric at infinity. This yields a natural notion of variational and linear stability for Ricci-flat ALF 4-metrics and lets us show that the conformally Kähler, non-hyperkähler examples are dynamically unstable along Ricci flow. We finally relate the sign of λ_ALF to positive relative mass statements for ALF metrics.Ricci flowRicci-flat & Calabi–YauKähler geometryStability & uniqueness https://neckpinch.com/papers/2201.05505/arxiv:2201.05505Fri, 14 Jan 2022 00:00:00 GMTJulius Baldauf, Dain Kim — This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.Ricci flowStability & uniqueness