Canonical diffeomorphisms of manifolds near spheres

For a given Riemannian manifold $(M^n, g)$ which is near standard sphere $(S^n, g_{round})$ in the Gromov-Hausdorff topology and satisfies $Rc \geq n-1$, it is known by Cheeger-Colding theory that $M$ is diffeomorphic to $S^n$. A diffeomorphism $\varphi: M \to S^n$ 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-H\"older estimate. We further show that this bi-H\"older 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.