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

Caleb Suan

doi:10.1515/crelle-2025-0051

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

Caleb Suan

Veröffentlicht/Copyright: 31. Juli 2025

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2025 Heft 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

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