https://neckpinch.com/authors/tristan-ozuch/New arXiv papers by Tristan Ozuch, 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/2510.21997arxiv: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://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/2410.16075arxiv:2410.16075Mon, 21 Oct 2024 00:00:00 GMTAlix Deruelle, Tristan Ozuch — In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension 4 with Einstein orbifolds as tangent flows at infinity. For instance, for any k∈ℕ₀, we obtain continuous families of non-isometric ancient Ricci flows on \#k(S²× S²) depending on a number of parameters growing linearly in k, and a family of half-PIC ancient Ricci flows on ℂP}²\#ℂP}². The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics.Ricci flowRicci solitonsAncient solutionsRicci-flat & Calabi–YauSingularitiesSphere theoremsStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2407.18438arxiv:2407.18438Fri, 26 Jul 2024 00:00:00 GMTIsaac M. Lopez, Tristan Ozuch — We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in SU(2) and mild decay at infinity are hyperkähler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's μ-functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized λ_ALE-functional related to a notion of weighted mass.Ricci flowRicci solitonsAncient solutionsRicci-flat & Calabi–YauKähler geometryScalar curvature https://arxiv.org/abs/2206.09198arxiv:2206.09198Sat, 18 Jun 2022 00:00:00 GMTJulius Baldauf, Tristan Ozuch — This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.Ricci flowBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2201.04475arxiv:2201.04475Wed, 12 Jan 2022 00:00:00 GMTJulius Baldauf, Tristan Ozuch — This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.Ricci flowScalar curvatureEigenvalues & spectrumPhysics