Paper · Jun 3, 2026

Calibration energy and mean curvature flow

Tatsuya Miura, Fabian Rupp

math.DGWider flowsarXiv:2606.04586v3
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As on arXiv

Abstract

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.

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Submitted
Jun 3, 2026
Last revised
Sep 1, 2026 (v3)
Subjects
math.DG, math.AP

Comments: 52 pages. Improved main results, which now cover coflat calibrations and singular initial data. Added new applications to dynamical rigidity of calibrated cones. Also slightly restructured. Comments welcome!

Mean curvature flowSurfacesWeak & singular flowsStability & uniquenessLong-time existence & convergence

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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 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 Weak-strong uniqueness for the mean curvature flow of double bubbles (Sebastian Hensel and Tim Laux, 2021)1Hensel 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)4Laux 2022 Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles (Marco Cicalese and Andrea Kubin, 2024)3Cicalese 2024 Stability of multiphase mean curvature flow beyond circular topology changes (Julian Fischer, Sebastian Hensel, Alice Marveggio et al., 2024)8Fischer 2024 Asymptotic circularity of immortal area-preserving curvature flows (Tatsuya Miura, 2024)7Miura 2024 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)5Li 2024 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)6Poiatti 2025 Dynamical stability of various convex graphical translators (Junyoung Park, 2026)2Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weak-strong uniqueness for the mean curvature flow of double bubbles

    Sebastian Hensel and Tim LauxWider flows

    Topics Surfaces, Weak & singular flows, Stability & uniqueness · Shared terms “calibrations”

  2. Dynamical stability of various convex graphical translators

    Junyoung ParkWider flows

    Topics Surfaces, Long-time existence & convergence, Stability & uniqueness · Shared terms “dynamical”

  3. Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

    Marco Cicalese and Andrea KubinWider flows

    Topics Surfaces, Weak & singular flows, Stability & uniqueness

  4. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “calibrations”, “calibrated”

  5. Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

    Xiang Li, Yong Luo and Jun SunWider flows

    Topics Surfaces, Stability & uniqueness · Shared terms “calibrated”, “rigidity theorem”, “Lagrangian”

  6. Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

    Andrea PoiattiWider flows

    Topics Weak & singular flows, Long-time existence & convergence, Stability & uniqueness · Shared terms “calibrated”, “calibrations”

  7. Asymptotic circularity of immortal area-preserving curvature flows

    Tatsuya MiuraWider flows

    Topics Long-time existence & convergence · Same author Tatsuya Miura · Shared terms “immortal”, “initial data”

  8. Stability of multiphase mean curvature flow beyond circular topology changes

    Julian Fischer, Sebastian Hensel, Alice Marveggio et al.Wider flows

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “flow beyond”, “initial data”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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