Paper · Nov 9, 2026

Lower bound of the Calabi functional and non-Archimedean K-energy

Tomoyuki Hisamoto

math.DGWider flowsarXiv:2611.11938
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As on arXiv

Abstract

We prove Donaldson's conjecture which describes the infimum of the Calabi functional by the normalized non-Archimedean K-energy. The proof makes use of proximal equation for the K-energy. To obtain approximate smooth solutions, a continuity path of prescribed scalar curvature equation is coupled with a version of Chen's -subgeodesic. It does not require a global smooth solution to the Calabi flow.

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Submitted
Nov 9, 2026
Subjects
math.DG, math.CV
Other curvature flowsKähler geometryScalar curvatureKähler–Einstein & K-stability

Its neighbourhood

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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.

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  1. Calabi flow with bounded scalar curvature (II)

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  2. Twisted Calabi functional and twisted Calabi flow

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  3. Hermitian Calabi functional in complexified orbits

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  4. Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold

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  5. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

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    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Shared terms “Calabi flow”, “continuity”, “Chen”

  6. Calabi flow with bounded scalar curvature

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    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Shared terms “Calabi flow”

  7. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Shared terms “Calabi flow”

  8. Calabi flow on polarized Kähler surfaces

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Shared terms “global smooth”, “Calabi”, “lower bound”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.