Canonical Diffeomorphisms of Manifolds Near Spheres
Canonical Diffeomorphisms of Manifolds Near Spheres
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DOI：
10.1007/s12220-023-01375-x
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
Bing Wang;Xinrui Zhao
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文献类型：
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作者：
Bing Wang;Xinrui Zhao
对于给定的黎曼流形，若M在Gromov-Hausdorff拓扑的标准球面附近，且满足，则由Cheeger-Colding理论可知，M与。在Cheeger和Colding（J Differ Geom 46（3）：406-480，1997）中使用Reifenberg方法构造了一个单同态。本文证明了一个理想的非同态可以正则地构造。设为（M，g）和的第一本征函数。则该映射提供一个自同态，并且满足一个一致的bi-Hölder估计.我们进一步证明了这个bi-Hölder估计是尖锐的，不能改进为bi-Lipschitz估计。我们的研究可以被认为是Colding的工作（Invent Math 124（1-3）：175-191，1996，Invent Math 124（1-3）：193-214，1996）和Petersen的工作（Invent Math 138（1）：1-21，1999）的延续。
For a given Riemannian manifoldwhich is near standard spherein the Gromov–Hausdorff topology and satisfies, it is known by Cheeger–Colding theory thatMis diffeomorphic to. A diffeomorphismwas constructed in Cheeger and Colding (J Differ Geom 46(3):406–480, 1997) using Reifenberg method. In this note, we show that a desired diffeomorphism can be constructed canonically. Letbe the first-eigenfunctions of (M,g) and. Then the mapprovides a diffeomorphism, andsatisfies a uniform bi-Hölder estimate. We further show that this bi-Hölder estimate is sharp and cannot be improved to a bi-Lipschitz estimate. Our study could be considered as a continuation of Colding’s works (Invent Math 124(1–3):175–191, 1996, Invent Math 124(1–3):193–214, 1996) and Petersen’s work (Invent Math 138(1):1–21, 1999).