https://neckpinch.com/authors/keaton-naff/New arXiv papers by Keaton Naff, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2511.15885arxiv:2511.15885Wed, 19 Nov 2025 00:00:00 GMTKeaton Naff, Tristan Ozuch — We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted L²-spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension 4. We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.Ricci solitonsKähler–Ricci solitonsSingularitiesKähler geometryStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2509.05470arxiv:2509.05470Fri, 05 Sep 2025 00:00:00 GMTKeaton Naff, Tristan Ozuch — We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly stable shrinking Ricci soliton. This offers new insights into the topological behavior of generic solutions to the Ricci flow in four dimensions: on top of reversing connected sums and handle surgeries, they should also undo complex blow-ups. The proof starts from an explicit description of the metric and develops a tensor harmonic analysis, adapted to its weighted Lichnerowicz Laplacian and based on its U(2)-invariance. It further exploits the Kähler structure of the blowdown shrinking soliton and insights from four-dimensional selfduality. The main difficulty is that the weighted Lichnerowicz Laplacian of the soliton admits a 9-dimensional set of eigentensors associated with nonnegative eigenvalues. We show that they correspond to the Ricci tensor and gauge transformations.Ricci flowRicci solitonsSingularitiesSurgery4-manifoldsKähler geometryStability & uniquenessEigenvalues & spectrum https://arxiv.org/abs/2504.17922arxiv:2504.17922Thu, 24 Apr 2025 00:00:00 GMTTang-Kai Lee, Keaton Naff, Jingze Zhu — We prove a parabolically scale-invariant variation of the planarity estimate in for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.Mean curvature flowRicci solitonsAncient solutions