1. Introduction
1.1. Background
Given an oriented Riemannian manifold M, how does the Lipschitz constant of a map
$M \to M$
control its degree? In all cases, if M is an n-manifold, an L-Lipschitz map
$M \to M$
multiplies n-dimensional volumes by at most
$L^n$
, and so its degree is at most
$L^n$
. In [Reference Gromov15, Ch. 2], Gromov studied the extent to which this estimate is sharp. For example, he showed that if M admits a sequence of self-maps
$f_k$
with
$$\begin{align*}\deg(f_k) \ge (1 - o(1)) \operatorname{\mathrm{Lip}}(f_k)^n,\end{align*}$$
then M must be flat [Reference Gromov15, 2.32]. He also asked the following question: for what M are there
$f_k$
with unbounded degree such that the ratio
$\operatorname {\mathrm {Lip}}(f_k)^n/\deg (f_k)$
is bounded [Reference Gromov15, 2.40(c)]? The answer to this modified question only depends on the topology of M. Gromov constructed such maps when M is a sphere or a product of spheres. He singled out
$(S^2 \times S^2) \mathbin{\#} (S^2 \times S^2)$
as a case in which he did not know whether such maps exist.
We now know that the answer for connected sums of copies of
$S^2 \times S^2$
or of
${\mathbb {C}} P^2$
is rather subtle. (The behavior is similar for both families.) Consider the manifold
$X_k = \#_k \mathbb CP^2$
. Volume considerations show that an L-Lipschitz self-map of any
$4$
-manifold has degree at most
$L^4$
. It is not difficult to construct an L-Lipschitz self-map of
$\mathbb CP^2$
with degree
$\sim L^4$
. When
$k=2$
or
$3$
, then [Reference Berdnikov and Manin3] shows that there are also L-Lipschitz self-maps of
$X_k$
with degree
$\sim L^4$
. But when
$k \ge 4$
, [Reference Berdnikov and Manin3] shows that every L-Lipschitz self-map of
$X_k$
has degree
$o(L^4)$
. Before this paper, the most efficient known maps had degree
$\sim L^3$
.
One of our goals in this paper is to give sharper quantitative estimates for the case
$k \ge 4$
. We will show that the maximal degree p lies in the range
$$\begin{align*}L^4(\log L)^{-4} \lesssim p \lesssim L^4(\log L)^{-1/2}.\end{align*}$$
This phase transition between
$k=3$
and
$k=4$
is an example of a broader phenomenon. Our second goal in the paper is to develop the general theory of this phenomenon.
For a given M, the maximally efficient relationship
$\operatorname {\mathrm {Lip}} f \sim (\deg f)^{1/n}$
may not be achievable for several reasons. For example, M may be inflexible, meaning that it does not have self-maps of degree
$>1$
. (Examples of inflexible simply connected manifolds are given in [Reference Arkowitz and Lupton2, Reference Crowley and Löh10, Reference Costoya and Viruel9, Reference Amann1].) Or it may be the case that any self-map of M of degree D multiplies some k-dimensional homology class by a factor greater than
$D^{k/n}$
, giving a stronger bound on the Lipschitz constant.
A compact manifold M is formal if it has a self-map
$M \to M$
which, for some p, induces multiplication by
$p^k$
on
$H_k(M;\mathbb R)$
, for every
$k \geq 1$
. This notion, first defined by Sullivan and coauthors in terms of rational homotopy theory, has played a role in many other geometric applications, starting with [Reference Deligne, Griffiths, Morgan and Sullivan11]. If M is a formal n-manifold, then obstructions to obtaining an L-Lipschitz map
$M \to M$
of degree
$L^n$
cannot come from measuring volumes of cycles. However, in [Reference Berdnikov and Manin3] it was shown that more subtle obstructions may exist. This motivates the definition of a scalable manifold to be one which has
$O(L)$
-Lipschitz self-maps of degree
$L^n$
. The paper [Reference Berdnikov and Manin3] shows that scalability is equivalent to several other conditions; most importantly, a manifold M (perhaps with boundary) is scalable if and only if there is a ring homomorphism
$H^*(M;{\mathbb {R}}) \to \Omega ^*(M)$
which realizes cohomology classes as differential forms representing them.
1.2. Main results
For nonscalable formal spaces, [Reference Berdnikov and Manin3] proves that any L-Lipschitz self-map has degree
$o(L^n)$
. Before this paper, the examples that had been constructed had degree
$O(L^{n-1})$
. In this paper, we gain a sharper quantitative understanding:
Theorem A. Let M be a formal, simply connected closed n-manifold which is not scalable. Then the maximal degree p of an L-Lipschitz map
$M \to M$
satisfies
$$\begin{align*}L^n(\log L)^{-\beta(M)} \lesssim p \lesssim L^n(\log L)^{-\alpha(M)},\end{align*}$$
where
$\beta (M) \geq \alpha (M)>0$
are constants depending only on the real cohomology ring of M.
For example, in the case of
$M=\#_k {\mathbb {C}} P^2$
,
$\beta (M)=4$
and
$\alpha (M)=1/2$
.
The lower bound of Theorem A generalizes to compact manifolds with boundary with a slightly more complicated statement (see Theorem 3.1).
We obtain a similar result for sizes of nullhomotopies of L-Lipschitz maps to a nonscalable formal space:
Theorem B. Let Y be a formal, simply connected compact Riemannian n-manifold (perhaps with boundary). Then for any finite simplicial complex X, any nullhomotopic L-Lipschitz map
$f:X \to Y$
is
$O(L(\log L)^{n-2})$
-Lipschitz nullhomotopic.
For scalable spaces, a linear bound was proved in [Reference Berdnikov and Manin3]; thus, this result is interesting mainly for nonscalable formal spaces. In contrast, in nonformal spaces, it is often impossible to do better than a bound of the form
$L^\alpha $
for some
$\alpha>1$
.
One of the main theorems of [Reference Berdnikov and Manin3] says that a manifold Y is scalable if and only if there is a ring homomorphism from
$H^*(Y; {\mathbb {R}})$
to
$\Omega ^*(Y)$
which takes each cohomology class to a differential form in that class. Because
$\Omega ^*(Y)$
is infinite-dimensional, this condition is not so easy to check. We verify the conjecture given in [Reference Berdnikov and Manin3] which states that scalability is equivalent to a simple homological criterion:
Theorem C. Let Y be a formal, simply connected compact Riemannian n-manifold (perhaps with boundary). Then Y is scalable if and only if there is an injective ring homomorphism
$$\begin{align*}h:H^*(Y; {\mathbb{R}}) \to \bigoplus_{i=1}^N \Lambda^*{\mathbb{R}}^{n_i}\end{align*}$$
for some integers
$n_1,\ldots ,n_N$
. In particular, if Y is a closed manifold, then it is scalable if and only if there is an injective ring homomorphism
$H^*(Y;\mathbb R) \to \Lambda ^*\mathbb R^n$
.
In particular, scalability is an invariant not only of rational but of real homotopy type.
Example 1.1. If M is an
$(n-1)$
-connected
$2n$
-manifold, then its real cohomology ring is completely described by the signature
$(k,\ell )$
of the bilinear form
$$\begin{align*}\smile:H^n(M;\mathbb R) \times H^n(M;\mathbb R) \to H^{2n}(M;\mathbb R).\end{align*}$$
Then M is scalable if and only if k and
$\ell $
are both at most
${2n \choose n}/2$
.
Theorem C is closely related to another idea studied by Gromov in [Reference Gromov15, 2.41]. For a closed n-manifold M, say a map
$f:\mathbb R^n \to M$
has positive asymptotic degree if
$$\begin{align*}\limsup_{R \to \infty} \frac{\int_{B_R(0)} f^*d\operatorname{\mathrm{vol}}_M}{R^n}=\delta>0.\end{align*}$$
Given an efficient self-map
$M \to M$
of high degree, you can zoom in and find a map of positive asymptotic degree on a large ball. If M is formal, then the converse also holds:
Theorem C′. Let M be a formal, simply connected closed n-manifold. Then a
$1$
-Lipschitz map
${f:\mathbb R^n \to M}$
of positive asymptotic degree exists if and only if M is scalable.
Remark 1.2. Gromov refers to manifolds with this property as elliptic, suggesting a connection with the notion of elliptic spaces from rational homotopy theory. However, this notion is not closely connected to scalability.
Question 1.3. Can a nonformal simply connected manifold be Gromov-elliptic?
Finally, we explore the behavior of nonformal manifolds:
Theorem D. Let M be a closed simply connected n-manifold which is not formal. Then either M is inflexible (has no self-maps of degree
$>1$
) or the maximal degree of an L-Lipschitz map
$M \to M$
is bounded by
$L^\alpha $
for some real number
$\alpha <n$
.
To see how the latter situation arises, consider the simplest example of a nonformal simply connected manifold, given in [Reference Félix, Oprea and Tanré13, p. 94]. This is the total space M of a fiber bundle
$S^3 \to M \to S^2 \times S^2$
obtained by pulling back the Hopf fibration
$S^3 \to S^7 \to S^4$
along the degree
$1$
map
$S^2 \times S^2 \to S^4$
.
A self-map of M is determined by its action on
$H^2(M) \cong \mathbb Z^2$
. This is because the generators of
$H^5(M)$
can be obtained from the generators of
$H^2(M)$
by taking Massey products (a higher cohomology operation) of order 3. An L-Lipschitz self-map takes the generators of
$H^5(M)$
to vectors of length
$O(L^5)$
, and therefore, it takes the generators of
$H^2(M)$
to vectors of length
$O(L^{5/3})$
. This means the degree of such a map is
$O(L^{20/3}) \prec L^7$
.
Something similar happens for any nonformal space: an alternate definition of formality is that a formal space has no nontrivial higher-order rational cohomology operations.
1.3. Proof ideas
The key idea behind Theorem A is that efficient self-maps of a formal but nonscalable space must behave nontrivially on many scales. We explain the intuition here.
In [Reference Berdnikov and Manin3], the
$o(L^n)$
upper bound for the degree of an L-Lipschitz map
$M \to M$
is obtained by looking at the induced pullbacks of differential forms representing cohomology classes of M and taking flat limits. To get the sharper upper bound of Theorem A, we analyze the same pullback forms using Fourier analysis – namely, Littlewood–Paley theory. These pullback forms can be decomposed into summands concentrated in different frequency ranges.
To start to get an idea how the proof works, first imagine that all the pullback forms are concentrated in a single frequency range. If the frequency range is high, then we got a lot of cancellation when we integrate the forms, leading to a nontrivial bound for the degree. If the frequency range is low, then we use the fact that M is not scalable to get a nontrivial bound for the degree – roughly speaking, if all the relevant forms were large and low frequency, we could use them to build a ring homomorphism from
$H^*(M; {\mathbb {R}})$
to
$\Omega ^*(M)$
.
In general, the pullback forms have contributions from many frequency ranges. We carefully break up the integral for the degree into pieces involving different frequency ranges, and we use the two ideas above to bound the pieces. It turns out that the interaction of different frequency ranges is important in this estimate. In the worst case, the forms have roughly equal contributions in every frequency range. Indeed, a self-map of M which comes close to the upper bound must have pieces in a wide range of frequencies (see Proposition 2.16 for a precise statement).
Let us see what such a self-map might look like in the case of
$M=\#_k {\mathbb {C}} P^2$
. We think of M as a CW complex with one
$0$
-cell,
$k 2$
-cells and one
$4$
-cell. We construct self-maps
${r}_\ell :M \to M$
which have degree
$2^{4\ell }$
on the top cell. We would like to arrange that
${r}_\ell $
has Lipschitz constant at most
$C \ell \cdot 2^{\ell }$
. A naive way to build a map
${r}_\ell $
of the right degree is to start with some
${r}_1$
and iterate it
$\ell $
times to get
${r}_\ell $
. In this case,
$\operatorname {\mathrm {Lip}}({r}_\ell ) \le \operatorname {\mathrm {Lip}}({r}_1)^\ell $
. However,
$\operatorname {\mathrm {Lip}}({r}_1)$
is strictly bigger than 2 (by [Reference Gromov15, 2.32], the Lipschitz constant could only be 2 if
$M = \#_k {\mathbb {C}} P^2$
had a flat metric). Therefore, the bound
$\operatorname {\mathrm {Lip}}({r}_1)^\ell $
is too big. By performing some optimization each time we iterate, we can bring
$\operatorname {\mathrm {Lip}}({r}_\ell )$
down to the target value.
We may build
${r}_1$
, which has degree
$16$
, as follows: the top cell
$e_4$
contains 16 cubical regions that each map homeomorphically, even homothetically, to the whole cell, whereas the area outside those cubical regions maps to the
$2$
-skeleton. To try to make this map efficient, we can arrange the cubical regions in a
$2 \times 2 \times 2 \times 2$
grid. But when we iterate this map many times, the regions that map homothetically to the
$4$
-cell become tiny, and most of the
$4$
-cell maps to the
$2$
-skeleton.
The main idea of the construction is that we can actually expand the homothetic regions so that they take up a much larger part of the cell, while compressing the parts that map to the
$2$
-skeleton to a thin layer. This has to do with the fact that self-maps of
$S^2$
of high degree are easy to produce and modify. In the end, each of the
$\ell $
iterations contributes a layer of roughly the same thickness, leading to an estimate of
$O(\ell \cdot 2^\ell )$
for the Lipschitz constant, or
$O(d^{1/4}\log d)$
in terms of the degree
$d=2^{4\ell }$
. See Figure 1 for a rough illustration.
Figure 1 Rescaling the ‘layers’ of the iterated map.
The proof of the lower bound of Theorem A is a straightforward generalization of this idea.
To end this introduction, we consider the Littlewood–Paley pieces of the differential forms from this map and from other maps we have discussed. For simplicity, let us first discuss a self-map
$S^2 \to S^2$
with degree
$2^{2p}$
and Lipschitz constant
$2^p$
. The pullback of the volume form is very repetitive, so that after averaging on scale
$2^{-p}$
, it becomes essentially constant. Therefore, the Littlewood–Paley pieces of the pullback are large at the highest frequency scale
$2^p$
and at frequency 1, but they can be very small at all the in-between frequencies.
The maps between scalable spaces constructed in [Reference Berdnikov and Manin3] have a similar Littlewood–Paley profile. These maps are highly regular ‘rescalings’. In fact, we prove Theorem C by building maps which are modeled on constant forms—the lowest possible frequency. Such maps are built on each cell and patched together using previous results from quantitative homotopy theory. The patching introduces high-frequency pieces, but there do not need to be any contributions from the intermediate frequencies.
The Littlewood–Paley decomposition for the self-map of
$\#_k {\mathbb {C}} P^2$
sketched above is very different. The outermost layer is dominated by very low-frequency terms (at scale around the diameter of the space) and very high-frequency terms (at scale
$\sim 2^{-\ell }$
). Similarly, the kth layer, which looks like the outermost layer but on a different scale, is dominated by terms at scale
$2^{-k}$
and
$2^{-\ell }$
. Overall, the map has pieces at every frequency range, as suggested by its fractal-like self-similarity.
1.4. Structure of the paper
Section 2 contains the Fourier-analytic proof of the upper bound of Theorem A; it is independent of the remainder of the paper. Section 3 discusses the corresponding lower bound and is likewise largely self-contained. Section 4 introduces some necessary results from rational and quantitative homotopy theory. In Section 5, we use this machinery to prove Theorems C and C′, and in Section 6, we use it to prove Theorem B. Finally, in Section 7, we discuss what our techniques can say about nonformal spaces, proving Theorem D as well as some complementary bounds.
2. Upper bounds on degree using Fourier analysis
In this section, we show the upper bound of Theorem A. To introduce the method, we first handle the case of a connected sum of
${\mathbb {C}} P^2$
s:
Theorem 2.1. Let
$X_k = \#_k {\mathbb {C}} P^2$
. Fix a metric g on
$X_k$
. Suppose that
$f: X_k \rightarrow X_k$
is L-Lipschitz. If
$k \ge 4$
, then
$$\begin{align*}\deg (f) \le C(k, g) L^4 (\log L)^{-1/2}.\end{align*}$$
We then use the same method to prove the general result:
Theorem 2.2. Suppose that M is a closed connected oriented n-manifold such that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
, and N is any closed oriented n-manifold. Then there is an
$\alpha (M)> 0$
so that for any metric g on M and
$g'$
on N and any L-Lipschitz map
$f: N \rightarrow M$
,
$$\begin{align*}\deg(f) \le C(M,g,N,g') L^n (\log L)^{- \alpha(M)}.\end{align*}$$
Note that by Theorem C, proved later in the paper, if M is simply connected and formal, then this condition holds if and only if M is not scalable. However, the theorem also holds for nonformal manifolds as well as those with nontrivial fundamental group.
A similar result also holds for many nonclosed domain manifolds. We give the proof for a unit ball, although it extends easily to any compact manifold with boundary:
Theorem 2.3. Suppose that M is a closed connected oriented n-manifold such that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
, and let
$\alpha (M)>0$
be as in the statement of Theorem 2.2. Let
$B^n \subseteq {\mathbb {R}}^n$
be the unit ball. Then for any metric g on M and any L-Lipschitz map
$f:B^n \to M$
,
$$\begin{align*}\int_{B^n} f^*d\operatorname{\mathrm{vol}}_M \leq C(M,g)L^n(\log L)^{-\alpha(M)}.\end{align*}$$
As discussed in the introduction, we prove these results by using Littlewood–Paley theory to divide the forms into pieces at different frequency ranges. In the first subsection, we review the tools from Littlewood–Paley theory that we need. In the second part, we prove Theorem 2.1. In the third part, we introduce the modifications needed to prove the more general estimate in Theorem 2.2.
2.1. Littlewood–Paley theory
If a denotes a differential form on
${\mathbb {R}}^d$
, then we can define its Fourier transform term by term. In other words, if I is a multi-index and
$ a = \sum _I a_I(x) dx^I$
, then
$$\begin{align*}\hat a := \sum_I \hat a_I dx^I.\end{align*}$$
To set up Littlewood–Paley theory, pick a partition of unity on Fourier space:
$$\begin{align*}\sum_{k \in {\mathbb{Z}}} \eta_k (\xi) := 1,\end{align*}$$
where
$\eta _k$
is supported in the annulus
$\operatorname {\mathrm {Ann}}_k := \{ \xi : 2^{k-1} \le |\xi | \le 2^{k+1} \}$
. We can also arrange that
$0 \le \eta _k \le 1$
and that
$\eta _k$
are smooth with appropriate bounds on their derivatives.
Then define
$$\begin{align*}P_k a := ( \eta_k \hat a)^{\vee},\end{align*}$$
where
$\vee $
denotes the inverse Fourier transform. We have
$a = \sum _{k \in {\mathbb {Z}}} P_k a$
, and we know that
$\widehat {P_k a} = \eta _k \hat a$
is supported in
$\operatorname {\mathrm {Ann}}_k$
.
We also write
$P_{\le k} a = \sum _{k' \le k} P_{k'} a$
, and
$\eta _{\le k} = \sum _{k' \le k} \eta _k$
, so
$P_{\le k} a = (\eta _{\le k} \hat a)^\vee $
.
We say that a form
$a = \sum _I a_I(x) dx^I$
is Schwartz if each function
$a_I(x)$
is Schwartz. A form a is Schwartz if and only if
$\hat a$
is Schwartz. Therefore, if a is Schwartz, then
$P_k a$
and
$P_{\le k} a$
are also Schwartz.
In this section, we review some estimates related to the
$P_k a$
. These results are proven using some inequalities about the inverse Fourier transform of smooth bump functions.
Lemma 2.4. Suppose that
$\eta (\omega )$
is a smooth function supported on a ball
$B\subset {\mathbb {R}}^d$
of radius 1 such that
-
•
$| \eta (\omega ) | \le A$
for all
$\omega $
. -
•
$| \partial _J \eta (\omega ) | \le A_N$
for all multi-indices J with
$|J| \le N$
.
Then
$$ \begin{align*} \lvert\eta^\vee(x)\rvert &\lesssim_d A \qquad\text{for every } x \in {\mathbb{R}}^d. \\ \lvert \eta^\vee(x) \rvert &\lesssim_d A_N \lvert x \rvert^{-N} \qquad \text{for every } x \in {\mathbb{R}}^d. \end{align*} $$
Therefore, if
$N> d$
,
$$\begin{align*}\lVert \eta^\vee \rVert_{L^1} \lesssim_d A + A_N.\end{align*}$$
Proof. For the first bound, we write
$$\begin{align*}|\eta^\vee(x)| = | {\textstyle\int \eta(\omega) e^{2 \pi i \omega x} d \omega} | \le {\textstyle\int} | \eta| \le |B| A.\end{align*}$$
For the second bound, we integrate by parts N times. For a given
$x \in {\mathbb {R}}^d$
, we choose a multi-index J with
$|J| =N$
and
$|x|^N \sim x^J$
. Then
$$\begin{align*}|\eta^\vee(x)| = \big\lvert {\textstyle\int \eta(\omega) e^{2 \pi i \omega x} d \omega} \big\rvert = \big\lvert{\textstyle\int \partial_J \eta (2 \pi i)^{-N} x^{-J} e^{2 \pi i \omega x} d \omega} \big\rvert \lesssim |x|^{-N} {\textstyle\int |\partial_J \eta|} \le \lvert x \rvert^{-N} \lvert B \rvert A_N.\end{align*}$$
To bound
$\int |\eta ^\vee (x)| dx$
, we use the first bound when
$|x| \le 1$
and the second bound when
$|x| \ge 1$
.
Lemma 2.5. Suppose that
$\eta (\omega )$
is a smooth function supported on a ball
$B\subset {\mathbb {R}}^d$
of radius R such that
-
•
$| \eta (\omega ) | \le A$
for all
$\omega $
. -
•
$| \partial _J \eta (\omega ) | \le A_N R^{-|J|}$
for all multi-indices J with
$|J| \le N$
.
Then
$$ \begin{align*} | \eta^\vee(x) | &\lesssim_d A R^d \qquad\textrm{ for every } x \in {\mathbb{R}}^d. \\ | \eta^\vee(x) | &\lesssim_d A_N R^d \lvert Rx \rvert^{-N} \qquad\textrm{ for every } x \in {\mathbb{R}}^d. \end{align*} $$
Therefore, if
$N> d$
,
$$\begin{align*}\| \eta^\vee \|_{L^1} \lesssim_d A + A_N.\end{align*}$$
Proof. The first two bounds follow from Lemma 2.4 by a change of variables. Alternatively, one can use the same method as in Lemma 2.4.
To bound
$\int |\eta ^\vee (x)| dx$
, we use the first bound when
$|x| \le 1/R$
and the second bound when
$|x| \ge 1/R$
.
Lemma 2.6. Suppose that
$\eta (\omega )$
is a smooth function supported on a ball
$B\subset {\mathbb {R}}^d$
of radius R such that
-
•
$| \eta (\omega ) | \le A$
for all
$\omega $
. -
•
$| \partial _J \eta (\omega ) | \le A_N R^{-|J|}$
for all multi-indices J with
$|J| \le N$
.
Write
$M f = \big ( \eta \hat f \big )^\vee $
. Then if
$N>d$
,
$$\begin{align*}\| M f \|_{L^p} \lesssim_d (A + A_N) \| f \|_{L^p}\text{ for every }1 \le p \le \infty.\end{align*}$$
Proof. We have
$Mf = f * \eta ^\vee $
. So
$\| M f \|_{L^p} \le \| f \|_{L^p} \| \eta ^\vee \|_{L^1}$
. Now apply the bound for
$\| \eta ^\vee \|_{L^1}$
from Lemma 2.5.
We apply these bounds to study the Littlewood–Paley projections
$P_k$
.
Lemma 2.7.
$\| \eta _k^\vee \|_{L^1} \lesssim 1$
uniformly in k.
$\| d \eta _k^\vee \|_{L^1} \lesssim 2^k$
uniformly in k.
Proof. We can first arrange that
$\eta _k(\omega ) = \eta _0(2^{-k} \omega )$
. Then the function
$\eta _k$
obeys the hypotheses of Lemma 2.5 with
$R = 2^k$
, with bounds that are uniform in k. Then Lemma 2.5 gives the estimate
$\| \eta _k^\vee \|_{L^1} \lesssim _d 1$
.
Next, we will show that
$\| \partial _j \eta _k^\vee \|_{L^1} \lesssim _d 2^k$
. This will imply
$\| d \eta _k^\vee \|_{L^1} \lesssim _d 2^k$
as desired.
The Fourier transform of
$\partial _j \eta _k^\vee $
is
$2 \pi i \omega _j \eta _k(\omega )$
. Notice that
$|\omega _j| \lesssim 2^k$
on
$\operatorname {\mathrm {Ann}}_k$
. We write
$$\begin{align*}2 \pi i \omega_j \eta_k = 2^k \cdot \underbrace{ 2 \pi i \frac{\omega_j}{2^k} \eta_k }_{\psi}.\end{align*}$$
The function
$\psi $
obeys the hypotheses of Lemma 2.5. Therefore,
$\| \psi ^\vee \|_{L^1} \lesssim _d 1.$
And so
$$\begin{align*}\| \partial_j \eta_k^\vee \|_{L^1} = 2^k \| \psi^\vee \|_{L^1} \lesssim 2^k. \\[-37pt]\end{align*}$$
Lemma 2.8.
$\| P_k a \|_{L^p} \le C \| a \|_{L^p}$
, for all k and all
$1 \le p \le \infty $
with a uniform constant C.
Proof.
$\|P_k a \|_{L^p} = \| \eta _k^\vee * a \|_{L^p} \le \| \eta _k^\vee \|_{L^1} \| a \|_{L^p}$
. Now
$\| \eta _k^\vee \|_{L^1}$
is bounded uniformly in k by Lemma 2.7.
Lemma 2.9. The projection operator
$P_k$
commutes with the exterior derivative d:
$$\begin{align*}d (P_k a) = P_k (da).\end{align*}$$
Proof. We can see this by taking the Fourier transform on both sides. The exterior derivative d becomes pointwise multiplication by a matrix on the Fourier side. The projection operator
$P_k$
becomes pointwise multiplication by the scalar
$\eta _k$
. These commute.
Lemma 2.10. Suppose that a is a Schwartz form on
${\mathbb {R}}^d$
with
$da = 0$
and with
$\hat a$
is supported in
$\operatorname {\mathrm {Ann}}_k:= \{ \xi : 2^{k-1} \le |\xi | \le 2^{k+1} \}$
. Then a has a primitive, which we denote
$\operatorname {\mathrm {Prim}}(a)$
, so that
-
•
$ d \operatorname {\mathrm {Prim}}(a) = a$
. (This is what the word ‘primitive’ means.) -
•
$\operatorname {\mathrm {Prim}}(a)$
is a Schwartz form. -
•
$ \lVert \operatorname {\mathrm {Prim}}(a)\rVert _{L^p} \le C 2^{-k} \lVert a \rVert _{L^p}$
for all
$1 \le p \le \infty $
, with a uniform constant C.
This is really the key property of frequency localized forms. The intuition is that
$\operatorname {\mathrm {Prim}}(a)$
is defined by integrating a, and the integral cancels at length scales larger than
$2^{-k}$
.
Before starting the proof, we make a quick remark about top-dimensional forms. If a is a d-form on
${\mathbb {R}}^d$
, then the condition
$da=0$
is automatic. In order for a to have a Schwartz primitive, we need to know that
$\int _{{\mathbb {R}}^d} a = 0$
. This fact is implied by our assumption that
$\hat a$
is supported in
$\operatorname {\mathrm {Ann}}_k$
because
$\int _{{\mathbb {R}}^d} a = \hat a(0) = 0$
.
Proof. First cover
$\operatorname {\mathrm {Ann}}_k$
with
$\sim 1$
balls B so that the radius of each ball is
$\sim 2^k$
and the distance from each ball to the origin is also
$\sim 2^k$
. Let
$\psi _B$
be a partition of unity:
$\sum _B \psi _B = 1$
on
$\operatorname {\mathrm {Ann}}_k$
and
$\psi _B$
is supported in B. Decompose
$a = \sum _B a_B$
where
$$\begin{align*}\hat a_B = \psi_B \hat a.\end{align*}$$
The form
$\hat a_B$
is smooth and supported in
$\operatorname {\mathrm {Ann}}_k \cup \operatorname {\mathrm {Ann}}_{k-1} \cup \operatorname {\mathrm {Ann}}_{k+1}$
. Just as in the proof of Lemma 2.9, it follows that
$d a_B = 0$
. Using Lemma 2.6,
$ \| a_B \|_{L^p} \le C \| a \|_{L^p}$
for all
$1 \le p \le \infty $
.
We will construct a primitive
$\operatorname {\mathrm {Prim}}(a_B)$
for each form
$a_B$
such that
-
•
$ d \operatorname {\mathrm {Prim}}(a_B) = a_B$
. -
•
$\operatorname {\mathrm {Prim}}(a_B)$
is a Schwartz form. -
•
$ \lVert \operatorname {\mathrm {Prim}}(a_B) \rVert _{L^p} \le C 2^{-k} \lVert a_B \rVert _{L^p}$
for all
$1 \le p \le \infty $
, with a uniform constant C.
Finally, we define
$\operatorname {\mathrm {Prim}}(a) = \sum _B \operatorname {\mathrm {Prim}}(a_B)$
. Since
$\operatorname {\mathrm {Prim}}(a_B)$
has the desired properties, it follows that
$\operatorname {\mathrm {Prim}}(a)$
does also.
Now we have to construct
$\operatorname {\mathrm {Prim}}(a_B)$
. For ease of notation, we will abbreviate
$a_B$
by a. We know that
$\hat a$
is supported on B. We can choose coordinates so that
$\omega _1 \sim 2^k$
on B.
We write the form a as
$$\begin{align*}\sum_I a_I(x) dx_I = \sum_{I = 1 \cup J} a_I(x) dx_1 \wedge dx_J + \sum_{1 \notin I} a_I dx_I.\end{align*}$$
We define the antiderivative
$\int a_I dx_1$
via the Fourier transform by the formula
$$ \begin{align} \widehat {\textstyle\int a_I dx_1} (\omega) = \frac{1}{2 \pi i \omega_1} \hat a_I(\omega). \end{align} $$
Since
$\omega _1> 0$
on B, and
$\hat a_I(\omega )$
is supported in B, the right-hand side is a smooth compactly supported function on Fourier space. Therefore,
$\int a_I dx_1$
is a Schwartz function on
${\mathbb {R}}^d$
. From (1), we can also check that
$$\begin{align*}\frac{\partial}{\partial x_1} \left( {\textstyle\int a_I dx_1} \right) = a_I.\end{align*}$$
We can also define
$\int a_I dx_1$
using definite integrals:
$$\begin{align*}\int a_I dx_1 (x_1, x_2, ..., x_d) = \int_{- \infty}^{x_1} a_I(\tilde x_1, x_2, ..., x_d) d \tilde x_1.\end{align*}$$
This definite integral formula is equivalent to (1). From the definite integral formula, it takes a little work to check that
$\int a_I dx_1$
is, in fact, a Schwartz function on
${\mathbb {R}}^d$
, although it is not that difficult. In our proof, we will only need (1).)
We now define
$$\begin{align*}\operatorname{\mathrm{Prim}}(a) = \sum_{I = 1 \cup J} ({\textstyle\int} a_I dx_1) dx_J.\end{align*}$$
This is a standard construction for primitives of forms which appears in the proof of the Poincaré lemma, cf. [Reference Bott and Tu6, p. 38]. We will check that
$d \operatorname {\mathrm {Prim}}(a) = a$
, following the same general method as in [Reference Bott and Tu6].
We first compute
$d( \int a_I dx_1)$
:
$$\begin{align*}d({\textstyle\int} a_I dx_1) = \partial_1 ({\textstyle\int} a_I dx_1) dx_1 + \sum_{j=2}^d \partial_j ({\textstyle\int} a_I dx_1) dx_j = a_I dx_1 + \sum_{j=2}^d {\textstyle\int} \partial_j a_I dx_1.\end{align*}$$
Now,
$$\begin{align*}d \operatorname{\mathrm{Prim}}(a) = \sum_{I = 1 \cup J} d ({\textstyle\int a_I dx_1}) dx_J = \sum_{I = 1 \cup J} a_I dx_1 \wedge dx_J + \sum_{I = 1 \cup J} \sum_{j=2}^d ({\textstyle\int \partial_j a_I dx_1}) dx_j \wedge dx_J.\end{align*}$$
The first term is
$\sum _{I = 1 \cup J} a_I dx_I$
. So we have to check that the second term is the rest of a. In other words, we want to show that
$$ \begin{align} \sum_{I = 1 \cup J} \sum_{j=2}^d ({\textstyle\int} \partial_j a_I dx_1) dx_j \wedge dx_J = \sum_{1 \notin I'} a_{I'} dx_{I'}. \end{align} $$
Since both forms are Schwartz, it suffices to check that
$\partial _1$
of both sides are equal:
$$ \begin{align} \sum_{I = 1 \cup J} \sum_{j=2}^d \partial_j a_I dx_j \wedge dx_J = \sum_{1 \notin I'} \partial_1 a_{I'} dx_{I'}. \end{align} $$
Since there is no 1 in J or j or
$I'$
, it suffices to check that
$dx_1$
wedged with both sides are equal:
$$ \begin{align} \sum_{I = 1 \cup J} \sum_{j=2}^d \partial_j a_I dx_1 \wedge dx_j \wedge dx_J = \sum_{1 \notin I'} \partial_1 a_{I'} dx_1 \wedge dx_{I'}. \end{align} $$
This, in turn, follows from
$da = 0$
.
To bound
$\operatorname {\mathrm {Prim}}(a)$
, the main point is that
$| \frac {1}{2 \pi i \omega _1}| \sim 2^{-k}$
on the ball B. Define
$\eta _B = 1 $
on B, and
$0 \le \eta _B \le 1$
and with
$\eta _B$
supported in a slightly larger ball
$\tilde B = 1.01 B$
. We can assume that
$\omega _1 \sim 2^k$
on
$\tilde B$
. Then
$$\begin{align*}\frac{1}{2 \pi i \omega_1} \hat a_I(\omega) = 2^{-k} \underbrace{\frac{1}{2 \pi i} \frac{2^k}{\omega_1} \eta_B}_{\tilde \eta_B} \hat a_I (\omega).\end{align*}$$
The function
$\tilde \eta _B$
is supported on
$\tilde B$
, and it obeys the bounds from Lemma 2.6. The lemma tells us that
$$\begin{align*}\| {\textstyle\int} a_I dx_1 \|_{L^p} = 2^{-k} \lVert\left( \tilde \eta_B \hat a_I \right)^\vee \rVert_{L^p} \le C 2^{-k} \lVert a_I \rVert_{L^p}.\end{align*}$$
Therefore,
$\lVert \operatorname {\mathrm {Prim}}(a)\rVert _{L^p} \le C 2^{-k} \lVert a \rVert _{L^p}$
as desired.
Lemma 2.11. For any function f,
$$\begin{align*}\sum_{k \in {\mathbb{Z}}} \| P_k f \|_{L^2}^2 \sim \| f \|_{L^2}^2.\end{align*}$$
Similarly, for any form a,
$$\begin{align*}\sum_{k \in {\mathbb{Z}}} \| P_k a \|_{L^2}^2 \sim \| a \|_{L^2}^2.\end{align*}$$
Proof. By the Plancherel theorem,
$$\begin{align*}\sum_{k \in {\mathbb{Z}}} \| P_k f \|_{L^2}^2 = \sum_{k \in {\mathbb{Z}}} \int_{{\mathbb{R}}^d} \big\lvert \widehat{P_k f} \big\rvert^2 = \sum_{k \in {\mathbb{Z}}} \int_{{\mathbb{R}}^d} \lvert\eta_k(\omega)\rvert^2 \lvert\hat f(\omega)\rvert^2 d \omega.\end{align*}$$
Now for every
$\omega $
,
$(1/10) \le \sum _{k \in {\mathbb {Z}}} \eta _k(\omega )^2 \le 1$
. This holds because
$\sum _{k \in {\mathbb {Z}}} \eta _k(\omega ) = 1$
and each
$\eta _k (\omega ) \ge 0$
, and each
$\omega $
lies in the support of
$\eta _k$
for at most 5 values of k. Therefore,
$$\begin{align*}\sum_{k \in {\mathbb{Z}}} \| P_k f \|_{L^2}^2 = \int_{{\mathbb{R}}^d} \bigg( \sum_{k \in {\mathbb{Z}}} \eta_k(\omega)^2 \bigg) |\hat f(\omega)|^2 d \omega \sim \int_{{\mathbb{R}}^d} |\hat f (\omega)|^2 d \omega = \int_{{\mathbb{R}}^d} |f(x)|^2 dx.\end{align*}$$
For a form
$a = \sum _{I} a_I(x) dx_I$
,
$P_k(a) = \sum _I P_k a_I(x) dx_I$
and
$\| a \|_{L^2}^2 := \sum _I \int |a_I(x)|^2 dx$
. So the case of forms follows from the case of functions.
Lemma 2.12. The Fourier support of
$P_{\le k} a_1 \wedge P_{\le k} a_2$
is contained in the ball of radius
$2^{k+2}$
around 0. Therefore,
$$\begin{align*}P_{\le k+3} \left( P_{\le k} a_1 \wedge P_{\le k} a_2 \right) = P_{\le k} a_1 \wedge P_{\le k} a_2.\end{align*}$$
Proof. The Fourier support of
$P_{\le k} a$
is contained in the ball
$B(2^{k+1}, 0)$
. For any functions f and g, the Fourier transform of
$fg$
is given by
$$\begin{align*}\widehat{fg}(\omega) = \hat f * \hat g(\omega) = \int \hat f(\tilde \omega) \hat g( \omega - \tilde \omega) d \tilde \omega.\end{align*}$$
If
$\hat f$
and
$\hat g$
are supported in
$B(2^{k+1}, 0)$
, then
$\widehat {fg}$
is supported in
$B( 2\cdot 2^{k+1}, 0)$
.
This argument also applies to wedge products of forms instead of products of functions, just by writing out the components of the forms. This shows that the Fourier transform of
$P_{\le k} a_1 \wedge P_{\le k} a_2$
is supported in
$B(2^{k+2}, 0)$
. Now,
$\eta _{\le k+3}(\omega )$
is identically 1 on this ball, and so
$$\begin{align*}P_{\le k+3} \left( P_{\le k} a_1 \wedge P_{\le k} a_2 \right) = P_{\le k} a_1 \wedge P_{\le k} a_2.\\[-42pt] \end{align*}$$
2.2. Bounds for connected sums of
$\mathbb CP^2$
s
2.2.1. Setup
In this section, we will prove Theorem 2.1. We recall the statement.
Theorem. Let
$X_k = ({\mathbb {C}} P^2)^{\# k}$
. Fix a metric g on
$X_k$
. Suppose that
$f: X_k \rightarrow X_k$
is L-Lipschitz. If
$k \ge 4$
, then
$$\begin{align*}\deg (f) \le C(k, g) L^4 (\log L)^{-1/2}.\end{align*}$$
Proof. Let
$u_i \in H^2(X_k; {\mathbb {R}})$
be a cohomology class dual to the ith copy of
${\mathbb {C}} P^1$
in
$X_k$
, for
$i = 1, \ldots , k$
. Let
$\alpha _i$
be a 2-form in the cohomology class
$u_i$
. We can assume that the
$\alpha _i$
have disjoint supports. For any i, we can write
$$ \begin{align} \deg (f) = \int_{X_k} f^* \alpha_i \wedge f^* \alpha_i. \end{align} $$
We will use Littlewood–Paley theory to estimate the right-hand side. Because Littlewood–Paley theory is by far nicest on
${\mathbb {R}}^d$
, we first switch to charts. Fix an atlas of charts for
$X_k$
: suppose that
$X_k = \cup U'$
, and
$\phi _U: U \rightarrow U'$
are parametrizations. Suppose that
$\sum _{U'} \psi _{U'} = 1$
is a partition of unity on
$X_k$
subordinate to these charts. Define
$\psi _U:{\mathbb {R}}^4 \to {\mathbb {R}}$
by
$$\begin{align*}\psi_U(x) = \begin{cases} \phi^{*} \psi_{U'}(x) & x \in U \\ 0 & x \notin U. \end{cases}\end{align*}$$
Now, we can extend
$\phi _U|_{\operatorname {\mathrm {supp}}(U)}$
to a smooth map
$\tilde \phi _U:\mathbb R^4 \to X_k$
, and we can do it so that
$\tilde \phi _U$
sends the complement of a compact set to a single point. Then define differential forms
$a_i$
on
${\mathbb {R}}^4$
by
$$ \begin{align} a_i = \tilde\phi_U^* f^*\alpha_i. \end{align} $$
(The forms
$a_i$
also implicitly depend on U.) Plugging this definition into (5), we get
$$ \begin{align} \deg (f) = \sum_U \int_{{\mathbb{R}}^4} \psi_U a_i \wedge a_i. \end{align} $$
We will bound each of these integrals.
Before going on, we discuss properties of the
$a_i$
. We made sure these forms are defined on all of
${\mathbb {R}}^4$
so that we can apply Littlewood–Paley theory. We have
$\| a_i \|_{L^\infty } \lesssim L^2$
. We also know that
$d a_i = 0$
. The form
$a_i$
is supported on a fixed ball, and so for every
$1 \le p \le \infty $
, we also have
$\| a_i \|_{L^p} \lesssim \| a_i \|_{L^\infty } \lesssim L^2$
.
2.2.2. Using that k is large
In this section, we prove a lemma that takes advantage of the fact that
$k \ge 4$
. This lemma is similar to a lemma in [Reference Berdnikov and Manin3].
Lemma 2.13. Suppose that
$k \ge 4$
and that
$b_1, \ldots , b_k$
are 2-forms on
${\mathbb {R}}^4$
. Then at each point x, we have
$$\begin{align*}| b_1 \wedge b_1 (x) | \le C \sum_{i \not= j} | b_i \wedge b_i - b_j \wedge b_j| + | b_i \wedge b_j|.\end{align*}$$
Proof. Suppose not. By scaling, we can assume that
$b_1 \wedge b_1(x) = dx_1 \wedge \cdots \wedge dx_4$
. Then we must have
$b_j \wedge b_j(x)$
is almost
$dx_1 \wedge \cdots \wedge dx_4$
for every j and
$b_i \wedge b_j(x)$
is almost zero for every
$i \neq j$
. Next, we will get a contradiction by considering the wedge product.
Let
$W: \Lambda ^2 {\mathbb {R}}^4 \times \Lambda ^2 {\mathbb {R}}^4 \rightarrow \Lambda ^4 {\mathbb {R}}^4$
be the quadratic form given by the wedge product. It has signature (3,3). Now, let
$B \subset \Lambda ^2 {\mathbb {R}}^4$
be the subspace spanned by
$b_1, \ldots , b_k$
. When we restrict W to the subspace B, we will check that it has signature
$(k,0)$
. Since
$k \ge 4$
, this gives the desired contradiction.
It remains to compute the signature of the quadratic form W restricted to B. This is isomorphic to the quadratic form
$(c_1, \ldots , c_k) \mapsto (\sum c_i b_i(x)) \wedge (\sum c_i b_i(x))$
. Expanding out the right-hand side, we get
$$\begin{align*}\sum_{i,j} c_i c_j b_i \wedge b_j.\end{align*}$$
Since
$b_i \wedge b_j$
is almost 0 for every
$i \neq j$
and
$b_i \wedge b_i$
is almost
$dx_1 \wedge \cdots \wedge dx_4$
for every i, we see that this form is almost
$$\begin{align*}(c_1, \ldots, c_k) \mapsto (c_1^2 + \cdots + c_k^2) dx_1 \wedge \cdots \wedge dx_4.\end{align*}$$
In particular, the form has signature
$(k,0)$
.
2.2.3. Relations in cohomology and low-frequency bounds
Let
$u_i \in H^2(X_k; {\mathbb {R}})$
be a cohomology class dual to the ith copy of
${\mathbb {C}} P^1$
in
$X_k$
, for
$i = 1, \ldots , k$
. Let
$\alpha _i$
be a 2-form in the cohomology class
$u_i$
.
We know that
$u_i \smile u_i - u_j \smile u_j = 0$
in
$H^4(X_k; {\mathbb {R}})$
. Therefore, the corresponding differential forms
$\alpha _i \wedge \alpha _i - \alpha _j \wedge \alpha _j $
are exact. Similarly, for
$i \neq j$
,
$u_i \smile u_j = 0$
, and so the forms
$\alpha _i \wedge \alpha _j$
are exact. Let
$\gamma _r$
be primitives for these forms. We have
$2{k \choose 2}$
exact forms total, and so r goes from
$1$
to
$2{k \choose 2}$
.
Define
$g_r = \phi ^* f^* \gamma _r$
. Since
$\gamma _r$
is a 3-form,
$$ \begin{align} \|g_r \|_{L^\infty} \lesssim L^3. \end{align} $$
Depending on r, we have
$dg_r = a_i \wedge a_i - a_j \wedge a_j$
or
$dg_r = a_i \wedge a_j$
with
$i \neq j$
.
The bound
$\| g_r \|_{L^\infty } \lesssim L^3$
gives extra information about
$a_i \wedge a_j$
. In particular, we get bounds on the low-frequency parts of
$a_i \wedge a_j$
.
Lemma 2.14. If
$i \neq j$
, then
$$ \begin{align*} \lVert P_{ k} ( a_i \wedge a_j) \rVert_{L^\infty} &\lesssim 2^k L^3 \\ \lVert P_{k} ( a_i \wedge a_i - a_j \wedge a_j) \rVert_{L^\infty} &\lesssim 2^k L^3. \end{align*} $$
The same bounds hold with
$P_{\le k}$
in place of
$P_k$
.
Notice that
$\| a_i \|_{L^\infty } \lesssim L^2$
, and so we have
$\| a_i \wedge a_j \|_{L^\infty } \lesssim L^4$
. But the low-frequency part of
$a_i \wedge a_j$
obeys a much stronger bound.
Proof. We write
$$\begin{align*}\left\lvert P_{k} (a_i \wedge a_j) (x) \right\rvert = \left\lvert \int \eta_{k}^\vee (y) a_i \wedge a_j (x-y) dy \right\rvert.\end{align*}$$
We now substitute in
$a_i \wedge a_j = d g_r$
and then integrate by parts:
$$\begin{align*}\left\lvert \int \eta_{k}^\vee (y) d g_r (x-y) dy \right\rvert = \left\lvert \int d \eta_{k}^\vee (y) g_r(x-y) dy \right\rvert.\end{align*}$$
Since
$\| g_r \|_{L^\infty } \lesssim L^3$
, and
$\int |d \eta _k^\vee | \lesssim 2^k$
by Lemma 2.7, our expression is bounded by
$$\begin{align*}\lesssim L^3 \int | d \eta_k^\vee| \lesssim 2^k L^3.\end{align*}$$
The same proof applies to
$ \| P_{k} ( a_i \wedge a_i - a_j \wedge a_j) \|_{L^\infty }$
and with
$P_{\le k}$
in place of
$P_k$
.
2.2.4. Toy case: all forms are low frequency
To illustrate how the tools we have developed work together, we now do a toy case of our main theorem: the case where all forms have low frequency.
Suppose that the forms
$a_i$
are all low-frequency:
$P_{\le 1} a_i = a_i$
for every i. It follows that the wedge products are also fairly low frequency:
$P_{\le 2} (a_i \wedge a_j) = a_i \wedge a_j$
for every
$i, j$
.
We can now bound
$\int \psi _U a_1 \wedge a_1$
using the tools we have developed. First, Lemma 2.13 tells us that
$$\begin{align*}\int \psi_U a_1 \wedge a_1 \le \int \psi_U |a_1 \wedge a_1| \le \sum_{i \neq j} \int \psi_U | a_i \wedge a_j| + \int \psi_U |a_i \wedge a_i - a_j \wedge a_j|.\end{align*}$$
We are discussing the low-frequency special case, where
$|a_i \wedge a_j| = | P_{\le 2} (a_i \wedge a_j) |$
. By Lemma 2.14, we have
$$\begin{align*}|a_i \wedge a_j| = | P_{\le 2} (a_i \wedge a_j) | \lesssim L^3.\end{align*}$$
Similarly,
$$\begin{align*}|a_i \wedge a_i - a_j \wedge a_j| = | P_{\le 2} (a_i \wedge a_i - a_j \wedge a_j) | \lesssim L^3.\end{align*}$$
Therefore,
$\int \psi _U a_1 \wedge a_1 \lesssim L^3$
, and so finally, we have
$\deg f \lesssim L^3$
.
If we have a weaker low-frequency assumption that
$P_{\le \bar \ell } a_i = a_i$
for every i, then the same argument shows that
$\deg f \lesssim 2^{\bar \ell } L^3$
. As long as the frequency range
$2^{\bar \ell }$
is significantly less than L, then we get a strong estimate. For instance, if
$2^{\bar \ell } = L^{.9}$
, then
$\deg f \le L^{3.9}$
.
2.2.5. Bounding high-frequency contributions
We use the Littlewood–Paley decomposition to write
$$\begin{align*}\int_{{\mathbb{R}}^d} \psi_U a_i \wedge a_i = \int_{{\mathbb{R}}^d} \psi_U \sum_{k \in {\mathbb{Z}}} P_k a_i \wedge \sum_{\ell \in {\mathbb{Z}}} P_\ell a_i.\end{align*}$$
We can bound each term on the right-hand side by using our primitive estimate, Lemma 2.10, and integration by parts:
$$ \begin{align*} \left\lvert\int_{{\mathbb{R}}^d} \psi_U P_k a_i \wedge P_\ell a_i \right\rvert &= \left\lvert\int_{{\mathbb{R}}^d} \psi_U P_k a_i \wedge d( \operatorname{\mathrm{Prim}}(P_\ell a_i) )\right\rvert \\ &= \left\lvert\int d \psi_U \wedge P_k a_i \wedge \operatorname{\mathrm{Prim}}( P_\ell a_i)\right\rvert \\ &\le \int \lvert d\psi_U \rvert \lvert P_k a_i \rvert \lvert\operatorname{\mathrm{Prim}} (P_\ell a_i)\rvert. \end{align*} $$
Now
$d \psi _U$
is a fixed
$C^\infty _{comp}$
form, and we have
$|P_k a_i| \lesssim L^2$
and
$| \operatorname {\mathrm {Prim}} P_\ell (a_i)| \lesssim 2^{-\ell } L^2$
. All together, we get the bound
$$ \begin{align} \left\lvert\int_{{\mathbb{R}}^d} \psi_U P_k a_i \wedge P_\ell a_i\right\rvert \lesssim 2^{-\ell} L^4. \end{align} $$
This shows that the high-frequency parts of
$a_i$
contribute little to the integral for the degree. By summing this geometric series of error terms, we see the following:
Lemma 2.15. For any frequency cutoff
$\bar \ell $
,
$$\begin{align*}\left\lvert\int_{{\mathbb{R}}^d} \psi_U a_i \wedge a_i\right\rvert \lesssim \left\lvert\int \psi_U P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_i \right\rvert + O(2^{-\bar \ell} L^4).\end{align*}$$
In particular, Lemma 2.15 allows us to resolve another toy case of our problem. If every form
$a_i$
is purely high-frequency, in the sense that
$P_{\le \bar \ell } a_i = 0$
, then Lemma 2.15 gives the bound
$\deg f \lesssim 2^{- \bar \ell } L^4$
. For instance, if
$2^{\bar \ell }$
is at least
$L^{1/10}$
, then we get a strong estimate:
$\deg f \lesssim L^{3.9}$
.
We now have strong bounds in two toy cases: the pure low-frequency case and the pure high-frequency case. We will prove bounds in the general case by combining these tools.
However, combining the tools is not completely straightforward. Based on the discussion above, it initially sounds like we might get a bound of the form
$\deg f \lesssim L^{4 - \beta }$
for some
$\beta> 0$
. But there are maps f with Lipschitz constant L and degree at least
$L^4 (\log L)^{-C}$
for some constant C. The forms coming from these maps crucially have signifinant contributions at all frequency levels.
2.2.6. Bounds in the general case
We begin by applying Lemma 2.15. For any frequency cutoff
$\bar \ell $
, the lemma tells us that
$$ \begin{align} \left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim \int \psi_U \left\lvert P_{\le \bar \ell} a_1 \wedge P_{\le \bar \ell} a_1\right\rvert + 2^{-\bar \ell} L^4. \end{align} $$
We will choose
$\bar \ell $
later, in the range
$2^{\bar \ell } \ge L^{1/10}$
. This guarantees that the last term is
$\lesssim L^{3.9}$
, which is much smaller than our goal.
To control the first term, we apply Lemma 2.13 with
$b_i = P_{\le \bar \ell } a_i(x)$
at each point x. Lemma 2.13 tells us that at each point,
$$\begin{align*}\left\lvert P_{\le \bar \ell} a_1 \wedge P_{\le \bar \ell} a_1 \right\rvert \lesssim \sum_{i \neq j} \left\lvert P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j \right\rvert + \left\lvert P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_i - P_{\le \bar \ell} a_j \wedge P_{\le \bar \ell} a_j\right\rvert.\end{align*}$$
Plugging into the integral, we get
$$\begin{align*}\int \psi_U | P_{\le \bar \ell} a_1 \wedge P_{\le \bar \ell} a_1 | \lesssim \sum_{i \neq j} \underbrace{\int \psi_U |P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j|}_{I} + \underbrace{\int \psi_U |P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_i - P_{\le \bar \ell} a_j \wedge P_{\le \bar \ell} a_j|}_{II}.\end{align*}$$
The two terms are similar to each other. We focus on the terms of type I first. The same arguments apply to type II.
The form
$P_{\le \bar \ell } a_i \wedge P_{\le \bar \ell } a_j$
looks a little bit like
$P_{\le \bar \ell } (a_i \wedge a_j)$
, which has strong bounds coming from Lemma 2.14. However, these forms are not equal to each other. We will examine the situation more carefully and find that
$$ \begin{align} P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j = P_{\le \bar \ell + 3} (a_i \wedge a_j) + \text{ additional terms}. \end{align} $$
The additional terms are crucial to our story – they actually make the largest contribution in our bound for the degree of f.
To work out the details of (11), we begin by doing the Littlewood–Paley expansion of
$a_i$
and
$a_j$
:
$$\begin{align*}a_i \wedge a_j = \sum_{k_1, k_2 \in {\mathbb{Z}}} P_{k_1} a_i \wedge P_{k_2} a_j.\end{align*}$$
Grouping the terms according to whether
$k_1$
or
$k_2$
is bigger, we get
$$ \begin{align} a_i \wedge a_j = P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j + \sum_{k_1 = \bar \ell+1}^\infty P_{k_1} a_i \wedge P_{\le k_1} a_j + \sum_{k_2 = \bar \ell+1}^\infty P_{< k_2} a_i \wedge P_{k_2} a_j. \end{align} $$
Note that the Fourier transform of
$P_{\le \bar \ell } a_i \wedge P_{\le \bar \ell } a_j$
is supported in
$|\omega | \le 4 \cdot 2^{\bar \ell }$
(cf. Lemma 2.12). Therefore,
$$\begin{align*}P_{\le \bar \ell + 3} ( P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j ) = P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j .\end{align*}$$
We apply
$P_{\le \bar \ell + 3}$
to both sides of (12) to get
$$ \begin{align} \begin{aligned} P_{\le \bar \ell +3} ( a_i \wedge a_j ) &= P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j \\ & \qquad {}+ \sum_{k_1 = \bar \ell+1}^\infty P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) + \sum_{k_2 = \bar \ell+1}^\infty P_{\le \bar \ell + 3} (P_{< k_2} a_i \wedge P_{k_2} a_j). \end{aligned} \end{align} $$
This gives us our fleshed out version of (11):
$$ \begin{align} \begin{aligned} P_{\le \bar \ell} a_i \wedge P_{\le \bar \ell} a_j &= \underbrace{P_{\le \bar \ell + 3} (a_i \wedge a_j)}_{\textrm{Term 1}} \\ & \qquad {} - \underbrace{ \sum_{k_1 = \bar \ell+1}^\infty P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) }_{\textrm{Term 2.1}}- \underbrace{\sum_{k_2 = \bar \ell+1}^\infty P_{\le \bar \ell + 3} (P_{< k_2} a_i \wedge P_{k_2} a_j)}_{\textrm{Term 2.2}}. \end{aligned} \end{align} $$
We want to bound
$\int \psi _U | P_{\le \bar \ell } a_i \wedge P_{\le \bar \ell } a_j |$
. We plug in (14), and then we have to bound the contributions of term 1, term 2.1 and term 2.2. The contribution of Term 1 is bounded using Lemma 2.14:
$$ \begin{align} \int \psi_U | P_{\le \bar \ell + 3} (a_i \wedge a_j)| \lesssim 2^{\bar \ell} L^3. \end{align} $$
We will choose
$\bar \ell $
in the range
$2^{\bar \ell } \le L^{9/10}$
, and so the right-hand side is
$\lesssim L^{3.9}$
, much smaller than our goal.
Terms 2.1 and 2.2 are similar, so we just explain Term 2.1. The contribution of Term 2.1 is at most
$$ \begin{align} \sum_{k_1 = \bar \ell+1}^\infty \int \psi_U \lvert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \rvert \le \sum_{k_1 = \bar \ell + 1}^\infty \lVert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \rVert_{L^1}. \end{align} $$
We start with a direct bound for this
$L^1$
norm. Lemma 2.8 gives
$$\begin{align*}\| P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \|_{L^1} \lesssim \| P_{k_1} a_i \wedge P_{\le k_1} a_j \|_{L^1} \le \| P_{k_1} a_i \|_{L^1} \| P_{\le k_1} a_j \|_{L^\infty}.\end{align*}$$
Now, Lemma 2.8 again gives
$\| P_{\le k_1} a_j \|_{L^\infty } \lesssim \| a_j \|_{L^\infty } \lesssim L^2$
. All together this gives
$$ \begin{align} \| P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \|_{L^1} \lesssim L^2 \| P_{k_1} a_i \|_{L^1}. \end{align} $$
If
$k_1 = \bar \ell $
, this is the best bound we know. But if
$k_1$
is much larger than
$\bar \ell $
, then we can get a better estimate by using the primitive of
$P_{k_1} a_i$
and integrating by parts.
$$\begin{align*}P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) = \eta_{\le \bar \ell + 3}^\vee * \left[ d \operatorname{\mathrm{Prim}} ( P_{k_1} a_i) P_{\le k_1} a_j \right].\end{align*}$$
Writing out what this means and integrating by parts, we get the following:
$$ \begin{align*} \left\lvert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) (x)\right\rvert &= \left\lvert \int \eta_{\le \bar \ell + 3}^\vee(y) ( d \operatorname{\mathrm{Prim}} ( P_{k_1} a_i))(x-y) \wedge P_{\le k_1} a_j (x-y) dy \right\rvert \\ &= \left\lvert \int d \eta_{\le \bar \ell + 3}^\vee(y) ( \operatorname{\mathrm{Prim}} ( P_{k_1} a_i))(x-y) \wedge P_{\le k_1} a_j (x-y) dy \right\rvert. \end{align*} $$
Therefore, we have a pointwise bound
$$\begin{align*}\left\lvert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \right\rvert \le \left\lvert d \eta_{\le \bar \ell + 3} * \left[ \operatorname{\mathrm{Prim}} (P_{k_1} a_i) \cdot P_{k_1} a_j \right]\right\rvert.\end{align*}$$
Taking
$L^1$
norms, we get
$$\begin{align*}\lVert P_{\le \bar \ell +3} (P_{k_1} a_i P_{\le k_1} a_j) \rVert_{L^1} \le \lVert d \eta_{\le \bar \ell + 3} \rVert_{L^1} \lVert \operatorname{\mathrm{Prim}} P_{k_1} a_i \rVert_{L^1} \lVert P_{k_1} a_j \rVert_{L^\infty}.\end{align*}$$
Now Lemma 2.7 gives
$ \| d \eta _{\le \bar \ell + 3} \|_{L^1} \lesssim 2^{\bar \ell }$
and Lemma 2.10 gives
$$\begin{align*}\lVert\operatorname{\mathrm{Prim}} P_{k_1} a_i \rVert_{L^1} \lesssim 2^{-k_1} \lVert P_{k_1} a_i \rVert_{L^1}.\end{align*}$$
We also know by Lemma 2.8 that
$\lVert P_{k_1} a_j \rVert _{L^\infty } \lesssim \lVert a_j \rVert _{L^\infty } \lesssim L^2$
. Putting these bounds together, we see that
$$ \begin{align} \lVert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \rVert_{L^1} \lesssim 2^{\bar \ell - k_1} L^2 \lVert P_{k_1} a_i \rVert_{L^1}. \end{align} $$
Returning to the contribution of Term 2.1 in (16), we have the bound
$$ \begin{align} \sum_{k_1 = \bar \ell+1}^\infty \int \psi_U \lvert P_{\le \bar \ell +3} (P_{k_1} a_i \wedge P_{\le k_1} a_j) \rvert \le \sum_{k_1 = \bar \ell + 1}^\infty 2^{\bar \ell - k_1} L^2 \lVert P_{k_1} a_i \rVert_{L^1}. \end{align} $$
Putting together our bounds for all the different terms, we get the following estimate for any choice of scale
$\bar \ell $
:
$$ \begin{align} \left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim 2^{- \bar \ell} L^4 + 2^{\bar \ell} L^3 + \sum_{k_1 = \bar \ell + 1}^\infty 2^{\bar \ell - k_1} L^2 \| P_{k_1} a_i \|_{L^1}. \end{align} $$
(On the right-hand side, the first term comes from high-frequency pieces, the next term comes from Term 1 and is bounded using the low-frequency method, and the final term comes from Terms 2.1 and 2.2. The fact that
$k \ge 4$
is used in the bound for Term 1.)
Let us pause to digest this bound. To begin, note that the first two terms,
$2^{- \bar \ell } L^4 + 2^{\bar \ell } L^3$
, can be made much smaller than
$L^4$
. For instance, we can choose
$\bar \ell $
so that
$2^{\bar \ell } = L^{1/2}$
, and then these first two terms give
$L^{3.5}$
. The final term is often the most important.
Now let us try to get some intuition about the last term. Because of the exponentially decaying factor
$2^{\bar \ell - k_1}$
, the final term comes mainly from
$k_1$
close to
$\bar \ell $
. If
$\| P_{k_1} a_i \|_{L^1}$
is very small for a range of
$k_1$
, then it is strategic for us to choose
$\bar \ell $
at the start of this range. This scenario could lead to a bound which is much stronger than
$L^4 (\log L)^{-1/2}$
– see Proposition 2.16 below. However, it may happen that
$\| P_{k_1} a_i \|_{L^1}$
are all roughly equal. This is actually the worst scenario from the point of view of Theorem 2.1. In this case, we can improve on the bound
$\| P_{k_1} a_i \|_{L^1} \lesssim \| a_i \|_{L^1} = L^2$
by using the orthogonality of the
$P_{k_1} a_i$
. By Cauchy–Schwarz,
$\| P_{k_1} a_i \|_{L^1} \lesssim \|P_{k_1} a_i \|_{L^2}$
, and
$\sum _{k_1} \| P_{k_1} a_i \|_{L^2}^2 \lesssim \| a_i \|_{L^2}^2 \lesssim L^4$
. If
$\|P_{k_1} a_i \|_{L^1}$
are all equal, then we can compute
$\| P_{k_1} a_i \|_{L^1} \lesssim L^2 (\log L)^{-1/2}$
. Plugging this into the last term, and summing the geometric series, the last term contributes
$L^4 (\log L)^{-1/2}$
.
We now finish the formal proof of Theorem 2.1. We apply our estimates for those
$\bar \ell $
satisfying
$L^{1/10} \le 2^{\bar \ell } \le L^{9/10}$
. The number of different
$\bar \ell $
in this range is
$\sim \log L$
. For each
$\bar \ell $
in this range, (20) gives
$$\begin{align*}\left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim L^{3.9} + \sum_{k_1 = \bar \ell + 1}^\infty 2^{\bar \ell - k_1} L^2 \| P_{k_1} a_i \|_{L^1}.\end{align*}$$
Summing the formula over all the
$\bar \ell $
in this range, we get
$$ \begin{align} \log L \left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim L^{3.91} + \sum_{L^{1/10} \le 2^{\bar \ell} \le L^{9/10}} \sum_{k_1 = \bar \ell + 1}^\infty 2^{\bar \ell - k_1} L^2 \| P_{k_1} a_i \|_{L^1}. \end{align} $$
In this sum, the terms with
$2^{k_1}> L$
can be bounded by
$L^{3.9}$
and absorbed into the first term. The remaining terms are
$$\begin{align*}\sum_{L^{1/10} \le 2^{k_1} \le L} \sum_{L^{1/10} \le 2^{\bar \ell} \le 2^{k_1}-1} 2^{\bar \ell - k_1} L^2 \| P_{k_1} a_i \|_{L^1} \lesssim \sum_{L^{1/10} \le 2^{k_1} \le L} L^2 \|P_{k_1} a_i \|_{L^1}.\end{align*}$$
Next, we want to use orthogonality from Lemma 2.11:
$\| a_i \|_{L^2}^2 \sim \sum _k \| P_k a_i \|_{L^2}^2$
. To get these
$L^2$
norms into play, we apply Cauchy–Schwarz. Since the
$a_i$
are supported in a fixed ball, and since the
$P_{k_1} a_i$
are rapidly decaying away from that ball, we have
$\| P_{k_1} a_i \|_{L^1} \lesssim \| P_{k_1} a_i \|_{L^2}$
. Since there are
$\sim \log L$
values of
$k_1$
in the range
$L^{1/10} \le 2^{k_1} \le L$
, we have
$$ \begin{align*} \sum_{L^{1/10} \le 2^{k_1} \le L} L^2 \|P_{k_1} a_i \|_{L^1} &\lesssim (\log L)^{1/2} L^2 \left( \sum_{L^{1/10} \le 2^{k_1} \le L} \|P_{k_1} a_i \|_{L^2}^2 \right)^{1/2} \\ &\lesssim (\log L)^{1/2} L^2 \| a_i \|_{L^2} \lesssim (\log L)^{1/2} L^4. \end{align*} $$
Plugging this back into (21), we see that
$$\begin{align*}\log L \left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim L^{3.91} + (\log L)^{1/2} L^4 \end{align*}$$
and so
$$\begin{align*}\left\lvert\int_{{\mathbb{R}}^d} \psi_U a_1 \wedge a_1\right\rvert \lesssim (\log L)^{-1/2} L^4.\end{align*}$$
But the degree of f is given by (7):
$$\begin{align*}\deg (f) = \sum_U \int \psi_U a_1 \wedge a_1 \lesssim (\log L)^{-1/2} L^4.\end{align*}$$
This finishes the proof of Theorem 2.1.
The bound (20) contains somewhat more information than Theorem 2.1. It also tells us that if the degree of f is close to
$L^4 (\log L)^{-1/2}$
, then the forms
$a_i$
must have contributions from essentially all frequency ranges. We make this precise in the following proposition.
Proposition 2.16. Suppose that
$k \ge 4$
. Suppose
$f: X_k \rightarrow X_k$
is L-Lipschitz. Let the forms
$a_i$
be as in (6), and fix
$0 < \beta _1 < \beta _2 < 1$
.
Suppose that for every chart and every i, and every
$k_1$
in the range
$L^{\beta _1} < 2^{k_1} < L^{\beta _2}$
,
$$ \begin{align} \lVert P_{k_1} a_i \rVert_{L^1} \le L^{2 - \gamma}. \end{align} $$
Then the degree of f is bounded by
$C(g) L^{4 - \eta }$
, where
$$\begin{align*}\eta = \min( \beta_1, \beta_2 - \beta_1, \gamma).\end{align*}$$
Proof. Recall that
$\| P_{k_1} a_i \|_{L^1} \lesssim \| a_i \|_{L^1} \lesssim L^2$
. The hypothesis (22) says that we have a stronger bound on
$\| P_{k_1} a_i \|_{L^1}$
when
$2^{k_1}$
lies in the range
$[L^{\beta _1}, L^{\beta _2}]$
.
To prove the bound, we plug all our hypotheses into the bound (20). That shows that the degree is bounded by
$$\begin{align*}L^{4 - \beta_1} + L^{3 + \beta_1} + \sum_{L^{\beta_1} \le 2^{k_1} \le L^{\beta_2}} L^{\beta_1} 2^{-k_1} L^2 L^{2 - \gamma} + \sum_{2^{k_1} \ge L^{\beta_2}} L^{\beta_1} 2^{-k_1} L^4.\end{align*}$$
Carrying out the geometric series and grouping terms finishes the proof.
2.3. General estimate
In this section, we prove theorem 2.2. We recall the statement.
Theorem. Suppose that M is a closed connected oriented n-manifold such that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
, and N is any closed oriented n-manifold. Then there exists
$\alpha (M)> 0$
so that for any metric g on M and
$g'$
on N and any map
$f: N \rightarrow M$
with
$\operatorname {\mathrm {Lip}}(f) = L$
,
$$\begin{align*}\deg(f) \le C(M,g,N,g') L^n (\log L)^{- \alpha(M)}.\end{align*}$$
Remark 2.17. The constant
$\alpha (M)$
depends only on the real cohomology algebra of M,
$H^*(M; {\mathbb {R}})$
.
Remark 2.18. Because the constant
$C(M,g)$
depends on g, it suffices to prove the estimate for any one metric g.
The main difference between the general situation in Theorem 2.2 and the special case
$X_k = ({\mathbb {C}} P^2)^{\# k}$
in Theorem 2.1 is to find the right analogue of Lemma 2.13. Lemma 2.13 takes advantage of the hypothesis that
$k \ge 4$
for
$X_k$
. Similarly, the following lemma takes advantage of the hypothesis that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
.
Lemma 2.19. Suppose that M is a closed connected oriented n-manifold such that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
. Then there exists an integer
$m(M)$
so that the following holds.
Let
$u_j \in H^{d_j} (M; {\mathbb {R}})$
be a set of generators for the cohomology algebra of M, including a generator
$u_{\operatorname {\mathrm {top}}} \in H^n(M; {\mathbb {R}})$
. Suppose that the relations of the cohomology algebra are given by
$R_r(u_1, \ldots , u_J) = 0$
.
Fix
$\beta _j \in \Lambda ^{d_j} {\mathbb {R}}^n$
for each
$j =1, \ldots , J$
such that
$|\beta _j| \le 1$
for each j and
$| R_r(\vec \beta )| \le {\epsilon }$
for each r. Then
$| \beta _{\operatorname {\mathrm {top}}} | \le C_M {\epsilon }^{\frac {1}{2m}}$
.
Proof. The tuple
$(\beta _1, \ldots , \beta _J)$
belongs to the space
$\prod _{j=1}^J \Lambda ^{d_j} {\mathbb {R}}^n$
, which is isomorphic to
${\mathbb {R}}^N$
. We can think of (each component of)
$\beta _j$
as a coordinate on this space, and we can think of
$R_r$
as a polynomial on this space. We let
$V(R_1, \ldots , R_k)$
be the set of
$\vec \beta $
where all the polynomials
$R_r$
vanish.
Each
$(\beta _1, \ldots , \beta _J) \in V(R_1, \ldots , R_k)$
corresponds to a homomorphism
$\phi : H^*(M; {\mathbb {R}}) \rightarrow \Lambda ^* {\mathbb {R}}^n$
with
$\beta _j = \phi (u_j)$
. By hypothesis, each such homomorphism is noninjective. By Poincaré duality, we have that each such homomorphism sends
$u_{\operatorname {\mathrm {top}}}$
to 0. Therefore,
$\beta _{\operatorname {\mathrm {top}}} = 0$
on
$V(R_1, \ldots , R_k)$
.
For any set
$X \subset {\mathbb {R}}^N$
, we let
$I(X)$
denote the ideal of polynomials
$f \in {\mathbb {R}}[\beta ]$
that vanish on X. So we see that
$\beta _{\operatorname {\mathrm {top}}} \in I (V (R_1, \ldots , R_k))$
. The structure of
$I( V( R_1, \ldots , R_k))$
is described by the real Nullstellensatz—cf. [Reference Marshall20, §2.3]:
Theorem 2.20 (Real Nullstellensatz)
A polynomial
$f \in {\mathbb {R}}[\beta ]$
lies in
$I (V (R_1, \ldots , R_k))$
if and only if there is an integer
$m \ge 1$
and polynomials
$g_i, h_r \in {\mathbb {R}}[\beta ]$
so that
$$\begin{align*}f^{2m} + g_1^2 + \ldots + g_s^2 = \sum_{r=1}^k h_r R_r.\end{align*}$$
By the real Nullstellensatz, we see that there is some integer m such that
$$\begin{align*}\beta_{\operatorname{\mathrm{top}}}^{2m} + g_1(\beta)^2 + \ldots + g_s(\beta)^2 = \sum_r h_r(\beta) R_r(\beta).\end{align*}$$
If we also know that
$|\beta _j| \le 1$
for every j and
$|R_r(\beta )| \le {\epsilon }$
for every r, then we see that
$$\begin{align*}\beta_{\operatorname{\mathrm{top}}}^{2m} \le C_M {\epsilon}.\end{align*}$$
Therefore,
$| \beta _{\operatorname {\mathrm {top}}} | \le C_M {\epsilon }^{\frac {1}{2m}}$
.
With this lemma, we can start the proof of the theorem. The ideas are the same. We just have to carry them out in a more general situation, with a little more notation.
Recall that
$u_j \in H^{d_j} (M; {\mathbb {R}})$
is a set of generators for the cohomology of M, with
$u_{\operatorname {\mathrm {top}}}$
the generator of
$H^n(M; {\mathbb {R}})$
. Suppose that the relations of the cohomology algebra are given by
$R_r(u_1, \ldots , u_J) = 0$
.
Choose
$\alpha _j$
to be a closed form on M in the cohomology class
$u_j$
. The cohomology class of
$R_r(\vec \alpha )$
is zero, so
$R_r(\vec \alpha )$
is exact. Choose a primitive:
$$\begin{align*}d \gamma_r = R_r(\vec\alpha).\end{align*}$$
Next, suppose that
$f: N \rightarrow M$
is an L-Lipschitz map. Cover N with charts
$U'$
, and let
$1 = \sum _{U'} \psi _{U'}$
be a partition of unity subordinate to the cover. Let
$\phi : U \rightarrow U'$
be a parametrization of
$U'$
, where
$U \subset {\mathbb {R}}^n$
, which extends to a smooth map
$\phi : {\mathbb {R}}^n \rightarrow M$
sending the complement of a large ball in
${\mathbb {R}}^n$
to a single point in M. Define a smooth compactly supported function
$$\begin{align*}\psi_U(x) = \begin{cases} \phi^{*} \psi_{U'}(x) & x \in U \\ 0 & x \notin U. \end{cases}\end{align*}$$
Define forms on
${\mathbb {R}}^n$
which correspond to the
$\alpha _j$
as follows:
$$\begin{align*}a_j := \frac{1}{L^{d_j}} \phi^* f^* \alpha_j.\end{align*}$$
With this normalization,
$\| a_j \|_{L^\infty } \lesssim 1$
and the
$a_j$
are smooth compactly supported differential forms. Then
$$ \begin{align} \deg(f) = L^n \sum_{U} \int_{{\mathbb{R}}^n} \psi_U a_{\operatorname{\mathrm{top}}}. \end{align} $$
Define forms on
${\mathbb {R}}^n$
which correspond to the
$\gamma _r$
as follows. If
$\gamma _r \in H^{d_r}(M; {\mathbb {R}})$
, then
$$\begin{align*}g_r := \frac{1}{L^{d(\gamma_r)+1}} \phi^* f^* \gamma_r.\end{align*}$$
The forms
$g_r$
are also smooth compactly supported differential forms. The power of L is chosen so that
$$\begin{align*}d g_r = R_r(a_j).\end{align*}$$
The power of L works out to make the forms
$g_r$
very small:
$$\begin{align*}\| g_r \|_{L^\infty} \lesssim L^{-1}.\end{align*}$$
This allows us to show that the low-frequency parts of the forms
$R_r(a)$
are small.
Lemma 2.21.
$\| P_{\le k} R_r(a) \|_{L^\infty } \lesssim 2^k L^{-1}$
.
Proof. We start by computing
$$\begin{align*}P_{\le k} R_r(a) (x) = \int_{{\mathbb{R}}^n} \eta_k^\vee(y) R_r(a) (x-y) dy = \int_{{\mathbb{R}}^n} \eta_k^\vee(y) dg_r (x-y) dy.\end{align*}$$
Now, we can integrate by parts to get
$$\begin{align*}\int_{{\mathbb{R}}^n} \eta_k^\vee(y) dg_r (x-y) dy = \int_{{\mathbb{R}}^n} d\eta_k^\vee(y) g_r (x-y) dy.\end{align*}$$
Taking norms and using
$\| g_r \|_{L^\infty } \lesssim L^{-1}$
, we see that
$$\begin{align*}| P_{\le k} R_r(a) (x) | \lesssim L^{-1} \int | d \eta_k^\vee| \lesssim 2^k L^{-1}.\\[-42pt] \end{align*}$$
We want to bound
$\int \psi _U a_{\operatorname {\mathrm {top}}}$
. We break this up into a low-frequency and high-frequency part at a frequency cutoff k which we will choose later. (Eventually we will average over many k.)
$$ \begin{align} \int \psi_U a_{\operatorname{\mathrm{top}}} = \underbrace{\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}}_{\textrm{low}} + \underbrace{\sum_{\ell> k} \int \psi_U P_\ell a_{\operatorname{\mathrm{top}}}}_{\textrm{high}}. \end{align} $$
For the high-frequency pieces in (24), we will find a small primitive and then integrate by parts. Lemma 2.10 tells us that
$P_{\ell } a_{\operatorname {\mathrm {top}}}$
has a primitive with
$$\begin{align*}\lVert \operatorname{\mathrm{Prim}} ( P_\ell a_{\operatorname{\mathrm{top}}}) \rVert_{L^\infty} \lesssim 2^{- \ell} \lVert P_{\ell} a_{\operatorname{\mathrm{top}}} \rVert_{L^\infty} \lesssim 2^{- \ell} \lVert a_{\operatorname{\mathrm{top}}} \rVert_{L^\infty} \lesssim 2^{- \ell}.\end{align*}$$
Then we can bound
$\int \psi _U P_{\ell } a_{\operatorname {\mathrm {top}}}$
by
$$\begin{align*}\int \psi_U P_{\ell} a_{\operatorname{\mathrm{top}}} = \int d \psi_U \operatorname{\mathrm{Prim}} (P_\ell a_{\operatorname{\mathrm{top}}}) \lesssim 2^{-\ell}.\end{align*}$$
We will choose k with
$2^k \ge L^{1/10}$
, and so the contribution of all the high-frequency parts is bounded by
$L^{-1/10}$
, which is much smaller than the bound we are aiming for.
For the low-frequency piece in (24)), we apply Lemma 2.19 to the forms
$P_{\le k} a_j$
. Since all these forms have norm
$\lesssim 1$
pointwise, the lemma gives us a pointwise bound
$$\begin{align*}|P_{\le k} a_{\operatorname{\mathrm{top}}}(x)| \lesssim \sum_r | R_r( P_{\le k} a)|^{\frac{1}{2m}}.\end{align*}$$
Integrating and using the Hölder inequality, we get the bound
$$ \begin{align} \int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}} \le \sum_r \int \psi_U | R_r(P_{\le k} a)|^{\frac{1}{2m}} \lesssim \sum_r \left( \int \psi_U |R_r(P_{\le k} a)| \right)^{\frac{1}{2m}}. \end{align} $$
In the Hölder step, in detail we wrote
$$ \begin{align*} \int \psi_U | R_r(P_{\le k} a)|^{\frac{1}{2m}} &= \int \psi_U^{\frac{2m-1}{2m}} \cdot \psi_U^{\frac{1}{2m}} |R_r(P_{\le k} a)|^{\frac{1}{2m}} \\ &\le \underbrace{\left( \int \psi_U \right)^{\frac{2m-1}{2m}}}_{\lesssim 1} \left( \int \psi_U |R_r(P_{\le k} a)| \right)^{\frac{1}{2m}}. \end{align*} $$
Now we have to bound each integral
$\int \psi _U |R_r( P_{\le k} a)|$
. Since
$\| a \|_{L^\infty } \lesssim 1$
, we get a bound
$\int \psi _U |R_r( P_{\le k} a)|\lesssim 1$
, and to prove our theorem, we need to beat this bound by a power of
$\log L$
, at least for some choice of k. The key input is the bound on the low-frequency part of
$R_r(a)$
: Lemma 2.21 tells us that
$\| P_{\le k} R_r(a) \|_{L^\infty } \le 2^{k} L^{-1}$
. Next, we have to relate
$R_r (P_{\le k} a)$
with
$P_{\le k} R_r(a)$
.
Remember that each
$R_r$
is a polynomial in the
$a_j$
. Each
$R_r(a_j)$
is a sum of terms of the form
$c a_{j_1} \wedge \cdots \wedge a_{j_P}$
. If we do a Littlewood–Paley decomposition of each
$a_j$
, we see that
$$ \begin{align} a_{j_1} \wedge \cdots \wedge a_{j_P} = \sum_{k_1, \ldots, k_P} P_{k_1} a_{j_1} \wedge \cdots \wedge P_{k_P} a_{j_P}. \end{align} $$
For each choice of
$k_1, \ldots , k_P$
, we write
$k_{\max } = \max _p k_p$
. We let
$p_{\max }$
be the value of p that maximizes
$k_p$
. If there is a tie, we let
$p_{\max }$
be the smallest p so that
$k_p = k_{\max }$
. We can now organize the sum on the right-hand side of (26) according to the value of
$k_{\max }$
and
$p_{\max }$
:
$$ \begin{align*} &\sum_{k_1, \ldots, k_P} P_{k_1} a_{j_1} \wedge \cdots \wedge P_{k_P} a_{j_P} = \sum_{k_{\max}} \sum_{p_{\max} = 1}^P P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge \\ &\quad{} \wedge P_{< k_{\max} } a_{j_{p_{\max} - 1}} \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge P_{\le k_{\max}} a_{j_{p_{\max}}+1} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}. \end{align*} $$
Similarly,
$$\begin{align*}P_{\le k} a_{j_1} \wedge \cdots \wedge P_{\le k} a_{j_P}= \sum_{k_{\max} \le k } \sum_{p_{\max} = 1}^P P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}.\end{align*}$$
Therefore,
$$ \begin{align*} &P_{\le k} a_{j_1} \wedge \cdots \wedge P_{\le k} a_{j_P} \\ &\quad= a_{j_1} \wedge \cdots \wedge a_{j_P} - \sum_{k_{\max}> k } \sum_{p_{\max} = 1}^P P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}. \end{align*} $$
This discussion applies to each monomial of
$R_r$
. Therefore,
$R_r(a)$
is equal to
$R_r( P_{\le k} a)$
plus a finite linear combination of terms of the form
$$ \begin{align} \sum_{k_{\max}> k } \sum_{p_{\max} = 1}^P P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}. \end{align} $$
Now, for a large constant c, we have
$P_{\le k+c} R_r( P_{\le k} a_j) = R_r( P_{\le k} a_j)$
. Therefore,
$ R_r( P_{\le k} a)$
is equal to
$P_{\le k+c} R_r(a) $
plus a finite linear combination of terms of the form
$$ \begin{align} \sum_{k_{\max}> k } \sum_{p_{\max} = 1}^P P_{\le k + c} \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right). \end{align} $$
In summary,
$$ \begin{align} R_r(P_{\le k} a) = P_{\le k+ c} R_r(a) + \text{ terms of the form}\ (28). \end{align} $$
The first term in (29) is controlled by Lemma 2.21:
$\| P_{\le k+ c} R_r(a) \|_{L^\infty } \lesssim 2^{k+c} L^{-1} \lesssim 2^k L^{-1}$
. We will choose k so that
$2^k \le L^{9/10}$
, so this term is bounded by
$L^{-1/10}$
, which is much smaller than our goal.
For each remaining term of type (28), we will again take a primitive and integrate by parts. We apply Lemma 2.10 to get a good primitive:
$P_{k_{\max }} a_{j_{p_{\max }}} = d \operatorname {\mathrm {Prim}} ( P_{k_{\max }} a_{j_{p_{\max }}} )$
, where
$\lVert \operatorname {\mathrm {Prim}} ( P_{k_{\max }} a_{j_{p_{\max }}} ) \rVert _{L^p} \lesssim 2^{-k_{\max }} \lVert P_{k_{\max }} a_{j_{p_{\max }}} \rVert _{L^p}$
for every
$1 \le p \le \infty $
. For each fixed choice of
$k_{\max }$
and
$p_{\max }$
, we write
$$ \begin{align*} &P_{\le k + c} \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right) \\ &\quad= \eta_{\le k + c}^\vee * \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge d \operatorname{\mathrm{Prim}}( P_{k_{\max}} a_{j_{p_{\max}}}) \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right) \\ &\quad= d \eta_{\le k + c}^\vee * \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge \operatorname{\mathrm{Prim}}( P_{k_{\max}} a_{j_{p_{\max}}}) \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right). \end{align*} $$
We now take the
$L^1$
norm of our term. Since
$\| a_j\|_{L^\infty }$
and
$\| P_{< k_{\max } a_j} \|_{L^\infty }$
are all
$\lesssim 1$
, we see that
$$ \begin{align*} &\big\lVert d \eta_{\le k + c}^\vee * \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge \operatorname{\mathrm{Prim}}( P_{k_{\max}} a_{j_{p_{\max}}}) \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right)\big\rVert_{L^1} \\ &\quad\lesssim \big\lVert d \eta_{\le k+ c}^\vee \big\rVert_{L^1} \big\lVert \operatorname{\mathrm{Prim}} ( P_{k_{\max}} a_{j_{p_{\max}}}) \big\rVert_{L^1} \lesssim 2^{k+c} 2^{-k_{\max}} \big\lVert P_{k_{\max}} a \big\rVert_{L^1}. \end{align*} $$
To summarize, we have proved the following bound on each summand of (28):
$$ \begin{align} \| P_{\le k + c} \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right) \|_{L^1} \lesssim 2^{k+c} 2^{-k_{\max}} \| P_{k_{\max}} a \|_{L^1}. \end{align} $$
Now, the
$L^1$
norm of each term of form (28) is bounded as follows:
$$ \begin{align*} &\bigg\lVert \sum_{k_{\max}> k } \sum_{p_{\max} = 1}^P P_{\le k + c} \left( P_{< k_{\max}} a_{j_1} \wedge \cdots \wedge P_{k_{\max}} a_{j_{p_{\max}}} \wedge \cdots \wedge P_{\le k_{\max}} a_{j_P}\right) \bigg\rVert_{L^1} \\ &\quad\lesssim \sum_{k_{\max}> k} 2^{k - k_{\max}} \| P_{k_{\max}} a \|_{L^1}. \end{align*} $$
We now have our bounds on all the terms, and we just have to put them together. Recall (25) tells us that
$$ \begin{align} \Big(\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}\Big)^{2m} \lesssim \sum_r \int \psi_U |R_r(P_{\le k} a)|. \end{align} $$
By (29), we can break up
$R_r(P_{\le k} a)$
into pieces:
$$ \begin{align*} R_r(P_{\le k} a) = P_{\le k+ c} R_r(a) + \text{terms of the form}\ (28). \end{align*} $$
We have now bounded each term on the right-hand side. Combining our bounds, we see that
$$\begin{align*}\Big(\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}\Big)^{2m} \lesssim \sum_r \int \psi_U |R_r(P_{\le k} a)| \lesssim 2^k L^{-1} + \sum_{k_{\max}> k} 2^{k-k_{\max}} \| P_{k_{\max}} a \|_{L^1}.\end{align*}$$
Let us pause to digest this bound. The first term
$2^k L^{-1}$
is very small as long as
$2^k \le L^{9/10}$
. In the second term, there is exponential decay for
$k_{\max }> k$
. Therefore, the main contribution on the right-hand side is when
$k_{\max } = k$
, which gives
$\| P_k a \|_{L^1}$
. For comparison, it would be straightforward to get an upper bound of
$\| a \|_{L^1} \lesssim 1$
. The upper bound
$\| P_k a \|_{L^1}$
is an improvement because it includes only one Littlewood–Paley piece of a. We can take advantage of this improvement by averaging over k and using orthogonality:
$\sum _k \| P_k a \|_{L^2}^2 \sim \| a \|_{L^2}^2$
. Now, we turn to the details of this estimate. We will sum over k in the range
$L^{1/10} \le 2^k \le L^{9/10}$
. There are
$\sim \log L$
different k in this range:
$$\begin{align*}\sum_{L^{1/10} \le 2^k \le L^{9/10}} \Big(\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}\Big)^{2m} \lesssim L^{-1/10} + \sum_{L^{1/10} \le 2^k \le L^{9/10}} \sum_{2^k < 2^{k_{\max}} < L} 2^{k-k_{\max}} \| P_{k_{\max}} a \|_{L^1}.\end{align*}$$
(Here, terms with
$2^{k_{\max }}> L$
are bounded by the
$L^{-1/10}$
term). Now, the last term is bounded by
$$\begin{align*}\sum_{L^{1/10} \le 2^k \le L^{9/10}} \sum_{2^k < 2^{k_{\max}} < L} 2^{k-k_{\max}} \| P_{k_{\max}} a \|_{L^1} \lesssim \sum_{L^{1/10} \le k_{\max} \le L} \| P_{k_{\max}} a \|_{L^1}.\end{align*}$$
The number of terms in this last sum is
$\sim \log L$
. Therefore, we can use the Cauchy–Schwarz inequality to get
$$\begin{align*}\sum_{L^{1/10} \le k_{\max} \le L} \| P_{k_{\max}} a \|_{L^1} \le (\log L)^{1/2} \bigg( \sum_{L^{1/10} \le 2^{k_{\max}} \le L} \| P_{k_{\max}} a \|_{L^1}^2 \bigg)^{1/2}.\end{align*}$$
Since a is supported on a fixed compact set, and
$P_{k_{\max }} a$
is essentially supported on that set, Cauchy–Schwarz gives
$\| P_{k_{\max }} a \|_{L^1} \lesssim \| P_{k_{\max }} a \|_{L^2}$
. Plugging this into the last term above gives
$$\begin{align*}(\log L)^{1/2} \bigg( \sum_{L^{1/10} \le 2^{k_{\max}} \le L} \| P_{k_{\max}} a \|_{L^2}^2 \bigg)^{1/2} \lesssim (\log L)^{1/2} \| a \|_{L^2}.\end{align*}$$
All together, we now have
$$\begin{align*}\sum_{L^{1/10} \le 2^k \le L^{9/10}} \Big(\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}} \Big)^{2m} \lesssim (\log L)^{1/2} \| a \|_{L^2}.\end{align*}$$
Since there are
$\sim \log L$
terms on the left-hand side, we can find one small term; that is, we can choose k in the range
$L^{1/10} \le 2^k \le L^{9/10}$
so that
$$\begin{align*}\Big(\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}\Big)^{2m} \lesssim (\log L)^{-1/2} \| a \|_{L^2} \lesssim (\log L)^{-1/2}.\end{align*}$$
Taking roots, we get
$\int \psi _U P_{\le k} a_{\operatorname {\mathrm {top}}} \lesssim (\log L)^{-\frac {1}{4m}}$
.
Recall that we broke up
$\int \psi _U a_{\operatorname {\mathrm {top}}}$
into low-frequency and high-frequency pieces in (24):
$$\begin{align*}\int \psi_U a_{\operatorname{\mathrm{top}}} = \underbrace{\int \psi_U P_{\le k} a_{\operatorname{\mathrm{top}}}}_{\textrm{low}} + \underbrace{\sum_{\ell> k} \int \psi_U P_\ell a_{\operatorname{\mathrm{top}}}}_{\textrm{high}}.\end{align*}$$
We previously showed that the high-frequency pieces are bounded by
$\lesssim 2^{-k}$
. Just now, we found k with
$L^{1/10} \le 2^k \le L^{9/10}$
, where the low-frequency piece has the bound
$\lesssim (\log L)^{- \frac {1}{4m}}$
. Therefore, the total is bounded:
$$\begin{align*}\int \psi_U a_{\operatorname{\mathrm{top}}} \lesssim (\log L)^{- \frac{1}{4m}}.\end{align*}$$
Recall from (23) that
$\deg f = L^n \sum _U \int \psi _U a_{\operatorname {\mathrm {top}}}$
, and so
$$\begin{align*}\deg f \lesssim L^n (\log L)^{- \frac{1}{4m}}.\end{align*}$$
This proves the theorem, with
$\alpha (m) = \frac {1}{4m}$
. The integer m came from the real Nullstellensatz, and it depended only on the cohomology ring
$H^*(M; {\mathbb {R}})$
.
2.3.1. Proof of Theorem 2.3
Finally, we describe the modifications needed to prove the result on the ball, which we restate here:
Theorem. Suppose that M is a closed connected oriented n-manifold such that
$H^*(M; {\mathbb {R}})$
does not embed into
$\Lambda ^* {\mathbb {R}}^n$
, and let
$\alpha (M)>0$
be as in the statement of Theorem 2.2. Let
$B^n \subseteq {\mathbb {R}}^n$
be the unit ball. Then for any metric g on M and any L-Lipschitz map
$f:B^n \to M$
,
$$\begin{align*}\int_{B^n} f^*d\operatorname{\mathrm{vol}}_M \leq C(M,g)L^n(\log L)^{-\alpha(M)}.\end{align*}$$
Proof. Our argument above already studies forms defined on a ball. The only difference is that above we study
$\int _{B^n} \psi f^*d\operatorname {\mathrm {vol}}_M$
, where
$\psi :B^n \to M$
is some function which decays to
$0$
at the boundary, whereas we now want to understand
$\int _{B^n} f^*d\operatorname {\mathrm {vol}}_M$
. To bridge the gap, we expand the domain. Define a function
$\tilde f:B_2(0) \to M$
on the ball of radius
$2$
by
$$\begin{align*}\tilde f(x)=\begin{cases} f(x) & \lVert x \rVert \leq 1 \\ f(x/\lVert x \rVert) & \lVert x \rVert>1. \end{cases}\end{align*}$$
If f is L-Lipschitz, this function is
$2L$
-Lipschitz. Moreover, since
$\tilde f$
has rank