Paper · Jul 12, 2016

Expanding Kähler-Ricci solitons coming out of Kähler cones

Ronan J. Conlon, Alix Deruelle

math.DGarXiv:1607.03546v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any and for any negative line bundle over a compact Kähler manifold , the total space of the vector bundle admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if . This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on with positive curvature operator on -forms is path-connected.

View PDF View HTML arXiv page
Submitted
Jul 12, 2016
Last revised
Sep 26, 2016 (v2)
Subjects
math.DG

Comments: 44 pages. Corrected several typos. Refined Theorem A and Corollary B

Ricci solitonsKähler–Ricci solitonsSphere theoremsKähler geometryWeak & singular flowsStability & uniquenessBoundaries & non-compact

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 2016, 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 Classification results for expanding and shrinking gradient Kähler-Ricci solitons (Ronan J. Conlon, Alix Deruelle and Song Sun, 2019)6Conlon 2019 Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones (Ronan J. Conlon and Alix Deruelle, 2020)5Conlon 2020 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)7Cifarelli 2022 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)8Bamler 2022 An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for t=0 (Ivin Babu, Ronan J. Conlon and Alix Deruelle, 2025)3Babu 2025 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)4Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)1Conlon 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)2Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “Kähler cone”, “gradient Kähler”, “resolution”

  2. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “gradient Kähler”, “soliton vector”, “vector field”

  3. An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for

    Ivin Babu, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “gradient Kähler”, “Kähler cone”, “asymptotically conical”

  4. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same author Ronan J. Conlon · Shared terms “expanding gradient”, “Kähler cone”, “gradient Kähler”

  5. Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “gradient Kähler”, “resolution”, “soliton vector”

  6. Classification results for expanding and shrinking gradient Kähler-Ricci solitons

    Ronan J. Conlon, Alix Deruelle and Song Sun

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “gradient Kähler”, “expanding gradient”, “Kähler cone”

  7. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “soliton vector”, “gradient Kähler”, “vector field”

  8. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Ronan J. Conlon, Alix Deruelle · Shared terms “gradient Kähler”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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