https://neckpinch.com/authors/akito-futaki/New arXiv papers by Akito Futaki, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2605.08796arxiv:2605.08796Sat, 09 May 2026 00:00:00 GMTAkito Futaki — The notion of weighted extremal Kähler metrics extends the classical notion of Calabi's extremal Kähler metrics, but includes many well-studied objects in Kähler geometry such as Kähler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal Kähler metrics.Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry https://arxiv.org/abs/2305.03544arxiv:2305.03544Fri, 05 May 2023 00:00:00 GMTAkito Futaki — We consider weighted solitons on Fano manifolds which include Kaehler-Ricci solitons, Mabuchi solitons and base metrics which induce Calabi-Yau cone metrics outside the zero sections of the canonical line bundles (Sasaki-Einstein metrics on the associated U(1)-bundles). In this paper, we give a condition for a weighted soliton on a Fano manifold M₀ to extend to weighted solitons on small deformations Mₜ of the Fano manifold M₀. More precisely, we show that all the members Mₜ of the Kuranishi family of a Fano manifold M₀ with a weighted soliton have weighted solitons if and only if the dimensions of T-equivariant automorphism groups of Mₜ are equal to that of M₀, and also if and only if the T-equivariant automorphism groups of Mₜ are all isomorphic to that of M₀, where the weight functions are defined on the moment polytope of the Hamiltonian T-action. This generalizes a result of Cao-Sun-Yau-Zhang for Kaehler-Einstein metrics.Ricci solitonsKähler–Ricci solitonsEinstein metricsRicci-flat & Calabi–YauKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry