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 $Rc \geq n-1$,itisknownbyCheeger-Coldingtheorythat$M$isdiffeomorphicto$S^n$. 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 $\tilde{f}=\frac{f}{\mathrm{\mid }f\mathrm{\mid }}:M\to S^n$ 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.