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$L^4$-Bound of the Transverse Ricci Curvature under the Sasaki-Ricci Flow

$L^4$-Bound of the Transverse Ricci Curvature under the Sasaki-Ricci Flow

Year:    2025

Author:    Shu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu

Journal of Mathematical Study, Vol. 58 (2025), Iss. 1 : pp. 38–61

Abstract

In this paper, we show that the uniform $L^4$-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian $(2n+1)$-manifold $M.$ Then we are able to study the structure of the limit space. As consequences, when $M$ is of dimension five and the space of leaves of the characteristic foliation is of type I, any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton and is trivial one if $M$ is transverse $K$-stable. Note that when the characteristic foliation is of type II, the same estimates hold along the conic Sasaki-Ricci flow.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v58n1.25.03

Journal of Mathematical Study, Vol. 58 (2025), Iss. 1 : pp. 38–61

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    Sasaki-Ricci flow Sasaki-Ricci soliton transverse Fano Sasakian manifold transverse Sasaki-Futaki invariant transverse $K$-stable Foliation singularities.

Author Details

Shu-Cheng Chang

Yingbo Han

Chien Lin

Chin-Tung Wu