Lemma 9.3.15A814C831E915

Proof. We use the properties of $\mathcal H^ s$-null sets recalled in section 5.11: subsets and countable unions of such sets, and their images under Lipschitz maps, are again $\mathcal H^ s$-null, and an $\mathcal H^ s$-null subset of $\mathbb R^ b$ with $s\leq b$ is Lebesgue null. Moreover, the image of a compact cube in $\mathbb R^ r$ under a Lipschitz map is $\mathcal H^ s$-null for $s>r$, by corollary 9.3.13(3), and a $C^1$ map is Lipschitz on every compact convex subset of its domain, by the mean value inequality.

(1) We argue by induction on the dimension of $N$. Since $N$ has at most countably many connected components, each of them open with a countable basis, we may assume that $N$ is connected, of dimension $n$. If $n=0$, then $F(N)$ is a point. Let $n\geq 1$, and let $r\leq k$ be the maximal rank of $\mathrm dF$ on $N$. The set $N_1$ of points where the rank is less than $r$ is the common zero set of the minors of size $r$ of the Jacobian matrix in local coordinates, so it is an analytic subset of $N$ different from $N$. Hence it is nowhere dense, and its germs, being proper subgerms of the irreducible manifold germs of dimension $n$ (lemma 9.3.12), have dimension at most $n-1$ by proposition 9.3.9. On the open set $N\setminus N_1$ the rank is constantly $r$.

Let $p\in N\setminus N_1$, and choose $\phi $, $\psi $ and $Q'\times Q''$ as in lemma 9.3.14, in a chart around $p$. Let $C'\subset Q'$ be a compact cube, a product of closed squares, containing the first $r$ coordinates of $\phi (p)$ in its interior, and let $V=\phi ^{-1}(\operatorname{Int}C'\times Q'')$. Then $F(V)\subset \psi (C'\times \{ 0\} )$, the image of a compact cube in $\mathbb R^{2r}$ under the restriction of $\psi $, which is Lipschitz there because $C'\times \{ 0\} $ is a compact convex subset of the convex domain of $\psi $. Hence $F(V)$ is $\mathcal H^ s$-null for $s>2r$, in particular for $s>2k$. Countably many such sets $V$ cover $N\setminus N_1$, because $N$ has a countable basis. By corollary 9.3.13(2), $N_1$ is the union of finitely many complex submanifolds of open subsets of $N$, of dimensions at most $n-1$; they have countable bases, and the restriction of $F$ to each of them has rank at most $k$. By the induction hypothesis their images are $\mathcal H^ s$-null for $s>2k$, and therefore so is $F(N)$.

(2) Suppose that $\dim (Y,y)\geq k+1$ for some $y\in Y$, and let $Y_1$ be a branch of $(Y,y)$ of dimension $e\geq k+1$. By corollary 9.3.6, $Y_1$ contains points of the covering locus of a parametrization of $Y_1$ outside the other branches of $(Y,y)$; near such a point $p$, the set $Y$ is a complex submanifold of dimension $e$, so it contains a complex submanifold $P$ of dimension $k+1$ through $p$, for instance a coordinate slice. The orthogonal projection $\Pi $ of $\mathbb C^ K$ onto the tangent space $T_ pP$, a complex subspace of dimension $k+1$, is $1$-Lipschitz, and its restriction to $P$ has invertible derivative at $p$. By the inverse function theorem, $\Pi (P)$ contains a nonempty open subset of $T_ pP\cong \mathbb R^{2k+2}$. On the other hand, $\Pi (P)\subset \Pi (Y)$ is $\mathcal H^{2k+2}$-null and hence Lebesgue null. This contradiction proves (2).

(3) By corollary 9.3.13(2), $A$ is the union of finitely many complex submanifolds $A_ l$ of open subsets of $N$, of dimensions at most $d$. Let $(V,\chi )$ be a holomorphic chart of $M$ around a point of $F(A)$, with values in $\mathbb C^ K$. The restriction of $\chi \circ F$ to the open subset $A_ l\cap F^{-1}(V)$ of $A_ l$ has rank at most $\dim A_ l\leq d$, so by (1) the set $\chi (F(A)\cap V)=\bigcup _ l\chi \circ F(A_ l\cap F^{-1}(V))$ is $\mathcal H^{2d+2}$-null. Near each of its points it is an analytic subset of an open subset of $\mathbb C^ K$, so (2) shows that $\dim (F(A),w)\leq d$ for $w\in F(A)\cap V$. $\square$