Canonical Diffeomorphisms of Manifolds Near Spheres

Abstract

Abstract For a given Riemannian manifold

$$(M^n, g)$$

(

M n

, g )

which is near standard sphere

$$(S^n, g_{round})$$

(

S n

,

g

round

)

in the Gromov–Hausdorff topology and satisfies

$$Rc \ge n-1$$

R c ≥ n - 1

, it is known by Cheeger–Colding theory that M is diffeomorphic to

$$S^n$$

S n

. A diffeomorphism

$$\varphi : M \rightarrow S^n$$

φ : M →

S n

was constructed in Cheeger and Colding (J Differ Geom 46(3):406–480, 1997) using Reifenberg method. In this note, we show that a desired diffeomorphism can be constructed canonically. Let

$$\{f_i\}_{i=1}^{n+1}$$

{

f i

}

i = 1

n + 1

be the first

$$(n+1)$$

( n + 1 )

-eigenfunctions of (M, g) and

$$f=(f_1, f_2, \ldots , f_{n+1})$$

f = (

f 1

,

f 2

, … ,

f

n + 1

)

. Then the map

$${\tilde{f}}=\frac{f}{|f|}: M \rightarrow S^n$$

f ~

=

f

| f |

: M →

S n

provides a diffeomorphism, and

$${\tilde{f}}$$

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 Colding’s works (Invent Math 124(1–3):175–191, 1996, Invent Math 124(1–3):193–214, 1996) and Petersen’s work (Invent Math 138(1):1–21, 1999).