8.1. Diffeomorphisms and Harmonic Radius [01Z4]

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8.1. Diffeomorphisms and Harmonic Radius

To control the diffeomorphism type of a manifold, or of part of a manifold, the basic tool one needs is to control the total number of coordinate charts, the number of domains these charts which can intersect a given chart and the change of coordinate maps between these charts in a suitably strong topology. This type of result has a long history, going back to [Ch1] in the context of bounded sectional curvature. In particular, control on the harmonic radius enables one to implement such an argument.

In this subsection we recall two theorems that will be used later. We refer the reader to the book [P] for proofs of these statements. The first theorem states that when two manifolds with harmonic radius bounded from below are sufficiently Gromov-Hausdorff close, then they must be diffeomorphic.

Theorem 8.1.

For every $\epsilon>0$ , there exists $\delta=\delta(n,\epsilon)$ , such that the following holds. If $M^{n}_{1},M^{n}_{2}$ are Riemannian manifolds and $U_{j}\subset M_{j}$ are subsets such that $r_{h}(x)>r>0$ for each $x\in U_{j}$ , and

d_{GH}(B_{r}(U_{1}),B_{r}(U_{2}))<\epsilon r\,,

then there exist open sets $B_{r/2}(U_{j})\subseteq U^{\prime}_{j}\subseteq B_{r}(U_{j})$ and a $C^{2}$ diffeomorphism $\Phi:U^{\prime}_{1}\to U^{\prime}_{2}$ , such that

\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}<\epsilon\,.

(8.1)

If we further assume $|{\rm Ric}_{M^{n}_{j}}|\leq n-1$ , $j=1,2$ , then $\Phi$ is in $C^{2,\alpha}\cap W^{3,q}$ for all $\alpha<1$ and $q<\infty$ , and in harmonic coordinates on $U^{\prime}_{1}$ we have

\displaystyle||g_{1}-\Phi^{*}g_{2}||_{C^{0}}+r^{1+\alpha}||\partial_{i}\Phi^{*}g_{2}||_{C^{\alpha}}+r^{2}||\partial_{i}\partial_{j}\Phi^{*}g_{2}||_{L^{q}}\leq C(n,\alpha,q)\epsilon\,.

(8.2)

The idea of the proof of Theorem 8.1 is to cover the set $U_{1}$ by harmonic charts $B_{r/2}(x_{j})$ of definite size, the intersection of whose domains also have a definite size or are empty and such that each chart domain intersects at most a definite number of distinct chart domains. By restricting the Gromov-Hausdorff map $f:U_{1}\to U_{2}$ to $U_{1}$ , and using that the image of each ball $f(B_{r}(x_{j}))$ lies in a harmonic coordinate chart of $U_{2}$ , we can construct a suitable smooth approximation of $f$ . Then using the estimates of the local charts one can see this smoothing of $f$ is the required diffeomorphism.

In a related direction, instead of trying to use the harmonic radius to directly to construct diffeomorphisms between nearby manifolds, we can use it to simply bound the number of diffeomorphism types of a space. Precisely, we have the following:

Theorem 8.2.

There exists $C=C(n,D)$ with the following property. Let $(M^{n},g)$ denote a Riemannian manifold and $U\subseteq M$ a subset such that $r_{h}(x)>r>0$ for all $x\in U$ and such that ${\rm diam}(U)\leq D\cdot r$ . Then there exists an open set $U^{\prime}$ with $T_{r/2}(U)\subseteq U^{\prime}\subseteq T_{r}(U)$ , such that $U^{\prime}$ has at most one of $C$ diffeomorphism types.

The idea of the proof of the above is that $U$ may be covered by a controlled number of harmonic charts $B_{r/2}(x_{j})$ with suitable control as above on the intersections of their domains. The geometry estimates on the charts automatically imply control over the transition functions between these charts. Hence there are a finite number of ways this finite collection of balls can be pasted together.