Abstract
For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an \(\epsilon \)-splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding n directional points by induction and stratified almost Gou–Gu Theorem. The other one is the error estimates of projections, which guarantee the n directional points we find really determine n different directions.
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Acknowledgements
We thank Zuoqin Wang for arranging our visit to University of Science and Technology of China, part of the work was done during the visit. Also we thank Zichang Liu for sharing his preprint [13] and some comments on the earlier version of this paper.
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Communicated by A. Neves.
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G. Xu: The first author was partially supported by Research was partially supported by Beijing Natural Science Foundation Z190003, NSFC 11771230 and NSFC 12141103.
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