https://neckpinch.com/authors/yury-ustinovskiy/New arXiv papers by Yury Ustinovskiy, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2509.01639arxiv:2509.01639Mon, 01 Sep 2025 00:00:00 GMTVestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets, Yury Ustinovskiy — We establish a local equivalence between toric steady Kähler-Ricci solitons and A-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are A-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.Ricci solitonsKähler–Ricci solitonsKähler geometry https://arxiv.org/abs/2404.14595arxiv:2404.14595Mon, 22 Apr 2024 00:00:00 GMTVestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy — Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and K-energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.Ricci solitonsKähler–Ricci solitonsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2211.09104arxiv:2211.09104Wed, 16 Nov 2022 00:00:00 GMTVestislav Apostolov, Xin Fu, Jeffrey Streets, Yury Ustinovskiy — We formulate an extension of the Calabi conjecture to the setting of generalized Kähler geometry. We show a transgression formula for the Bismut Ricci curvature in this setting, which requires a new local Goto/Kodaira-Spencer deformation result, and use it to show that solutions of the generalized Calabi-Yau equation on compact manifolds are classically Kähler, Calabi-Yau, and furthermore unique in their generalized Kähler class. We show that the generalized Kähler-Ricci flow is naturally adapted to this conjecture, and exhibit a number of a priori estimates and monotonicity formulas which suggest global existence and convergence. For initial data in the generalized Kähler class of a Kähler Calabi-Yau structure we prove the flow exists globally and converges to this unique fixed point. This has applications to understanding the space of generalized Kähler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKähler manifolds. In the case of commuting-type generalized Kähler structures we establish global existence and convergence with arbitrary initial data to a Kähler, Calabi-Yau metric, which yields a new d d^c-lemma for these structures.Ricci flowKähler–Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence https://arxiv.org/abs/2206.01196arxiv:2206.01196Thu, 02 Jun 2022 00:00:00 GMTYury Ustinovskiy — In this paper we study the gradient steady Kähler-Ricci soliton metrics on non-compact toric manifolds. We show that the orbit space of the free locus of such a manifold carries a natural Hessian structure with a nonnegative Bakry-Émery tensor. We generalize Calabi's classical rigidity result and use this to prove that any complete mathbf Tⁿ-invariant gradient steady Kähler-Ricci soliton with a free torus action must be a flat (ℂ^*)ⁿ.Ricci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCDStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2109.10295arxiv:2109.10295Tue, 21 Sep 2021 00:00:00 GMTVestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy — Under broad hypotheses we derive a scalar reduction of the generalized Kähler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the classification in complex dimension 2.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness