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 $\phi :M\to S^n$ was constructed by Cheeger and Colding 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 $f=(f_1,f_2,\cdots ,f_{n+1})$. Then the map $\tilde{f}=\frac{f}{|f|}:M\to S^n$ 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 the previous works of Colding and Petersen.