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For a given Riemannian manifold $(M^n, g)$ which is near standard sphere$(S^n, g_{round})$ in the Gromov-Hausdorff topology and satisfies $Rc \geqn-1$, 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 Coldingusing Reifenberg method. In this note, we show that a desired diffeomorphismcan be constructed canonically. Let $\{f_i\}_{i=1}^{n+1}$ be the first$(n+1)$-eigenfunctions of $(M, g)$ and $f=(f_1, f_2, \cdots, f_{n+1})$. Thenthe 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 thisbi-H\"older estimate is sharp and cannot be improved to a bi-Lipschitzestimate. Our study could be considered as a continuation of the previous worksof Colding and Petersen.

Canonical diffeomorphisms of manifolds near spheres