Paper · Dec 4, 2024

The Kähler-Ricci soliton on bounded pseudoconvex domains

Zehao Sha

math.CVarXiv:2412.03345v4
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Abstract

In this paper, we study Kähler-Ricci solitons on bounded pseudoconvex domains in with boundary. Under suitable assumptions, we prove that such solitons must be Kähler-Einstein. Building on Huang and Xiao's resolution of Cheng's conjecture, we further establish an analogous result for Bergman Kähler-Ricci solitons. Several model domains are presented to illustrate our results.

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Submitted
Dec 4, 2024
Last revised
Apr 14, 2026 (v4)
Subjects
math.CV, math.DG

Comments: 19 pages. More details added

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stability

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Invariant Szegő metric on strongly pseudoconvex domains

    Anjali Bhatnagar and Jiliang Fan

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “pseudoconvex domain”, “Einstein”

  2. On the Bergman metric of Cartan-Hartogs domains

    Andrea Loi, Roberto Mossa and Fabio ZUddas

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

  3. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa

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

  4. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

  5. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al.

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

  6. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

  7. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

  8. An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

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.