Canonical Diffeomorphisms of Manifolds Near Spheres

Abstract

For a given Riemannian manifold $(M^n, g)$ which is near standard sphere $(S^n, g_{round})$ in the Gromov–Hausdorff topology and satisfies $Rc \ge n-1$, it is known by Cheeger–Colding theory that M is diffeomorphic to $S^n$. A diffeomorphism $\varphi : M \rightarrow S^n$ 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 $f=(f_1, f_2, \ldots , f_{n+1})$. Then the map ${\tilde{f}}=\frac{f}{|f|}: M \rightarrow S^n$ provides a diffeomorphism, and ${\tilde{f}}$ 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).