Canonical Diffeomorphisms of Manifolds Near Spheres

• Published in: The Journal of geometric analysis 2023-09, Vol.33 (9), Article 304
• Main Authors: Wang, Bing, Zhao, Xinrui
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Eigenvectors; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Isomorphism; Mathematics; Mathematics and Statistics; Riemann manifold; Topology
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01375-x

For a given Riemannian manifold ( M n , g ) which is near standard sphere ( S n , g round ) in the Gromov–Hausdorff topology and satisfies R c ≥ n - 1 , it is known by Cheeger–Colding theory that M is diffeomorphic to S n . A diffeomorphism φ : M → S n was 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. Let { f i } i = 1 n + 1 be the first ( n + 1 ) -eigenfunctions of ( M ,  g ) and f = ( f 1 , f 2 , … , f n + 1 ) . Then the map f ~ = f | f | : M → S n provides a diffeomorphism, and f ~ satisfies 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).