Paper · May 9, 2026

On weighted extremal Kähler metrics

Akito Futaki

math.DGarXiv:2605.08796
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Abstract

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.

Comments: Submitted to the Proceedings of ICCM 2025

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper On canonical radial Kaehler metrics (Andrea Loi, Filippo Salis and Fabio Zuddas, 2021)6Loi 2021 Weighted extremal metrics on blowups (Michael Hallam, 2023)2Hallam 2023 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)3Futaki 2023 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)4Hättig 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)8Li 2023 On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five (Shu-Cheng Chang, Fengjiang Li, Chien Lin et al., 2024)7Chang 2024 The weighted Hermite–Einstein equation (Michael Hallam and Abdellah Lahdili, 2024)5Hallam 2024 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)1Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Shared terms “extremal Kähler”

  2. Weighted extremal metrics on blowups

    Michael Hallam PDF

    Topics Sasakian & contact geometry, Kähler–Einstein & K-stability, Kähler–Ricci solitons · Shared terms “weighted extremal”, “Sasaki”

  3. Deformations of Fano manifolds with weighted solitons

    Akito Futaki PDF

    Topics Sasakian & contact geometry, Kähler–Einstein & K-stability, Kähler–Ricci solitons · Same author Akito Futaki · Shared terms “Sasaki Einstein”, “Calabi”

  4. Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

    Daniel Hättig, Jürgen Hausen and Hendrik Süß PDF

    Topics Sasakian & contact geometry, Kähler–Einstein & K-stability, Kähler–Ricci solitons · Shared terms “Sasaki Einstein”

  5. The weighted Hermite–Einstein equation

    Michael Hallam and Abdellah Lahdili PDF

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Einstein metrics · Shared terms “Sasaki”

  6. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Shared terms “extremal”

  7. On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

    Shu-Cheng Chang, Fengjiang Li, Chien Lin et al. PDF

    Topics Sasakian & contact geometry, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Sasaki Einstein”

  8. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows PDF

    Topics Kähler–Einstein & K-stability, Kähler geometry · Shared terms “extremal Kähler”, “Calabi”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.