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 $$Rc \ge n-1$$ , it known by Cheeger–Colding theory that M diffeomorphic to $$S^n$$ . A diffeomorphism $$\varphi : \rightarrow S^n$$ was constructed Cheeger Colding (J Differ Geom 46(3):406–480, 1997) using Reifenberg method. In this note, we show desired can be canonically. Let $$\{f_i\}_{i=1}^{n+1}$$ first $$(n+1)$$ -eigenfunctions of (M, g) $$f=(f_1, f_2, \ldots f_{n+1})$$ Then map $${\tilde{f}}=\frac{f}{|f|}: provides diffeomorphism, $${\tilde{f}}$$ uniform bi-Hölder estimate. We further estimate sharp cannot improved bi-Lipschitz Our study could considered as continuation Colding’s works (Invent Math 124(1–3):175–191, 1996, Invent 124(1–3):193–214, 1996) Petersen’s work 138(1):1–21, 1999).