On the long time behaviour of the conical Kähler–Ricci flows

Xiuxiong Chen; Yuanqi Wang

doi:10.1515/crelle-2015-0103

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On the long time behaviour of the conical Kähler–Ricci flows

Xiuxiong Chen and Yuanqi Wang

Published/Copyright: March 1, 2016

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 744

Abstract

We prove that the conical Kähler–Ricci flows introduced in [11] exist for all time t∈[0,+∞). These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class C1,β is negative or zero, the corresponding conical Kähler–Ricci flows converge to Kähler–Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville-type theorem for Kähler–Ricci flat metrics (which are defined over ℂn) with conical singularities.

A Appendix: Trudinger’s Harnack inequality

The proof of Lemma A.1 can be seen in [22], [15], and [18]. We only elaborate briefly how to use the weak conical condition to obtain Euclidean energy estimates.

Lemma A.1 (Trudinger’s Harnack inequality).

Suppose ω is a weak conical metric. For any ball B⁢(R) with radius R (centered at any point), suppose

(A.1)Δω⁢u≤0⁢in the weak sense in⁢B⁢(R),u∈W1,2⁢[B⁢(R)]∩C2⁢[B⁢(R)∖D],

and u is nonnegative almost everywhere. Then for all 0<p<nn-1, we have

R-2⁢np⁢|u|Lp⁢B⁢(R2)≤C⁢(p)⁢infB⁢(R4)⁡u,

where the Lp-norm is with respect to the volume form of ω. Consequently, suppose

(A.2)Δω⁢u=0 𝑖𝑛⁢B⁢(R).

and u attains interior maximum or minimum. Then u is a constant over B⁢(R).

Proof.

Without loss of generality, we assume R=1. Consider u¯=u+k with k>0. Let v=1u¯. Then v is a positive weak-subsolution to Δω⁢v≥0. Since ω is a weak conical metric, the following holds by definition:

(A.3)gEC≤ω≤C⁢gE over ⁢B⁢(1)∖D,

where gE is the Euclidean metric in the polar coordinates. Let φ=η2⁢vp, by using the Cauchy–Schwarz inequality, we obtain

(A.4)2⁢p(p+1)2⁢∫B⁢(1)η2⁢|∇ω⁡vp+12|2⁢ωn≤4p⁢∫B⁢(1)|∇ω⁡η|2⁢vp+1⁢ωn.

By (A.3) and (A.4),

(A.5)2⁢p(p+1)2⁢∫B⁢(1)η2⁢|∇E⁡vp+12|2⁢𝑑volE≤Cp⁢∫B⁢(1)|∇E⁡η|2⁢vp+1⁢𝑑volE.

Then from [18, Chapter 4, Theorem 1.1] we deduce for any p>0 that

supB⁢(14)⁡v≤C⁢(p)⁢|v|Lp⁢B⁢(12).

Then

(A.6)infB⁢(14)⁡u¯≥(∫B⁢(12)u¯-p)-1p=(∫B⁢(12)u¯p⁢𝑑volE)1p(∫B⁢(12)u¯p⁢𝑑volE)1p⁢(∫B⁢(12)u¯-p⁢𝑑volE)1p.

Notice (A.6) is a Euclidean estimate. By using the easy approach in (A.3), (A.4), (A.5) to find Euclidean energy estimates, and running the other parts of the proof in [18, 15], the proof is complete. ∎

Acknowledgements

The first named author wishes to thank Kai Zheng, Chengjiang Yao for helpful discussions on the maximal principle over conical settings. He is also grateful to Xi Zhang for sharing his insight on weak conical Kähler–Ricci flow. The second named author wish to thank Professor S. K. Donaldson for communications on earlier versions of this work. The second author is grateful to Professors Xianzhe Dai, Guofang Wei, and Rugang Ye for their interest in this work and their continuous support. He also would like to thank Yuan Yuan for related discussions on complex analysis. Both authors would like to thank Song Sun, Kai Zheng, and Haozhao Li for carefully reading earlier versions of this paper and related discussions.

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Received: 2015-02-02

Revised: 2015-09-11

Published Online: 2016-03-01

Published in Print: 2018-11-01

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