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

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.

Keywords

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

Acknowledgements

The first author would like to express his gratitude to Professor Xiaochun Rong for long-term inspirations such as the Gromov-Hausdorff compactness. Part of this project was carried out during the third named author’s visit to the NCTS, whose support he would like to express the gratitude for the warm hospitality.

The first author is supported in part by Funds of the Mathematical Science Research Center of Chongqing University of Technology (Grant No. 0625199005). The fourth au-thor partially supported in part by the NSTC of Taiwan. The second author partially sup-ported by an NSFC 11971415, NSF of Henan Province 252300421497, and Nanhu Scholars Program for Young Scholars of Xinyang Normal University. The third author partially supported by the Science and Technology Research Program of Chongqing Municipal Education. Commission (Grant No. KJQN202201165).

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