Canonical Diffeomorphisms of Manifolds Near Spheres
Name
12220_2023_1375_ReferencePDF.pdf
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Author(s)
Wang, Bing
Zhao, Xinrui
Date Issued
July 12, 2023
Publisher
Springer US
Citation
The Journal of Geometric Analysis. 2023 Jul 12;33(9):304
Version
Author's final manuscript
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).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s12220-023-01375-x
