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Anomaly flow: Shi-type estimates and long-time existence

  • Caleb Suan ORCID logo EMAIL logo
Published/Copyright: July 31, 2025
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 826

Abstract

We consider the long-time existence of the anomaly flow on a compact complex 3-fold with general slope parameter α ′ . In particular, we obtain integral Shi-type estimates for the flow by adapting an integration-by-parts type argument instead of the usual maximum principle techniques. Following this, we prescribe a sufficient smallness condition on α ′ in order to extend the flow on [ 0 , τ ) to [ 0 , τ + ϵ ) .

A Identities for Hermitian metrics and Chern connections

In this appendix, we list some useful identities that will be used often. The conformally balanced condition d ⁢ ( ∥ Ω ∥ ω ⁢ ω 2 ) = 0 is equivalent to

T i = ∇ i log ∥ Ω ∥ ω , T ̄ j ̄ = ∇ ̄ j ̄ log ∥ Ω ∥ ω .

As such, we see that

∇ ( 1 2 ⁢ ∥ Ω ∥ ω ) = − ( 1 2 ⁢ ∥ Ω ∥ ω ) ⋅ T , ∇ ̄ ⁢ ( 1 2 ⁢ ∥ Ω ∥ ω ) = − ( 1 2 ⁢ ∥ Ω ∥ ω ) ⋅ T ̄ .

Repeated application of the above yields the following:

(A.1) ∇ m ∇ ̄ l ⁢ ( 1 2 ⁢ ∥ Ω ∥ ω ) = ( 1 2 ⁢ ∥ Ω ∥ ω ) ⁢ ∑ i 1 + ⋯ + i r + ( r − s ) = m j 1 + ⋯ + j s + s = l ∇ i 1 ∇ ̄ j 1 ⁢ T ̄ ∗ ⋯ ∗ ∇ i s ∇ ̄ j s ⁢ T ̄ ∗ ∇ i s + 1 T ∗ ⋯ ∗ ∇ i r T .

We also note the general commutator identities: for a generic tensor 𝐴,

(A.2) ∇ m ∇ ̄ l ⁢ ( Δ R ⁢ A ) = Δ R ⁢ ( ∇ m ∇ ̄ l ⁢ A ) + ∑ i = 0 m ∑ j = 0 l ( ∇ m − i ∇ ̄ l − j ⁢ A ) ∗ ( ∇ i ∇ ̄ j ⁢ Rm ) + ∑ i = 0 m ∑ j = 0 l ( ∇ m − i ∇ ̄ l + 1 − j ⁢ A ) ∗ ( ∇ i ∇ ̄ j ⁢ T ) + ∑ i = 0 m ∑ j = 0 l ( ∇ m + 1 − i ∇ ̄ l − j ⁢ A ) ∗ ( ∇ i ∇ ̄ j ⁢ T ̄ ) ,

and

(A.3) ∇ ̄ l ⁢ ∇ m A = ∑ r = 0 min ⁡ ( m , l ) ∑ i 0 + ⋯ + i r = m − r j 0 + ⋯ + j r = l − r ( ∇ i 0 ∇ ̄ j 0 ⁢ A ) ∗ ( ∇ i 1 ∇ ̄ j 1 ⁢ Rm ) ∗ ( ∇ i r ∇ ̄ j r ⁢ Rm ) .

Lastly, we have the divergence theorem for the Chern connection

(A.4) ∫ X ∇ i V i = ∫ X T i ⋅ V i and ∫ X ∇ ̄ j ̄ ⁢ V j ̄ = ∫ X T ̄ j ̄ ⋅ V j ̄ .

Acknowledgements

The author thanks his supervisor Sébastien Picard for suggesting this topic of study and also for many helpful conversations and discussions. The author also thanks the referee for their comments and also for their careful reading of the paper.

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Received: 2024-11-27
Revised: 2025-06-30
Published Online: 2025-07-31
Published in Print: 2025-09-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. No compact split limit Ricci flow of type II from the blow-down
  3. Deriving Perelman’s entropy from Colding’s monotonic volume
  4. Gravitational instantons with 𝑆1 symmetry
  5. Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
  6. On the Rapoport--Zink space for GU(2, 4) over a ramified prime
  7. Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
  8. On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
  9. Anomaly flow: Shi-type estimates and long-time existence
https://doi.org/10.1515/crelle-2025-0051

Articles in the same Issue

  1. Frontmatter
  2. No compact split limit Ricci flow of type II from the blow-down
  3. Deriving Perelman’s entropy from Colding’s monotonic volume
  4. Gravitational instantons with 𝑆1 symmetry
  5. Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
  6. On the Rapoport--Zink space for GU(2, 4) over a ramified prime
  7. Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
  8. On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
  9. Anomaly flow: Shi-type estimates and long-time existence
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