Topological invariants and Holomorphic Mappings

It is an elegant and effective technique in Riemannian geometry to consider the minimum-length curve in each nontrivial free homotopy class of closed curves in a compact Riemannian manifold. Such a minimum-length curve always exists and is a smooth closed geodesic. This idea is, for example, the basic step in proving that a compact, orientable manifold of even dimension with everywhere positive sectional curvature is simply connected (cf. [21] p. 172, Theorem 26). If the Riemannian manifold is not compact, then a minimal length curve may not exist, but it remains of interest to consider the infimum of the lengths of closed curves in the free homotopy class and also the infimum over all nontrivial free homotopy classes as well.

The purpose of this paper is to examine this general idea in the context of complex manifolds and also to consider the corresponding possibilities for the situation of maps of the $k$-sphere which are homotopically nontrivial, with length replaced by a suitable idea of $k$-dimensional measure.

For Riemann surfaces which are covered by the unit disc $\Delta$—that is, all Riemann surfaces except $\mathbb{C},\mathbb{C}\setminus\{0\},\mathbb{C}\cup\{\infty\}$, and compact surfaces of genus 1—the situation becomes one of Riemannian metrics: If $\pi\colon\Delta\to M$ is a holomorphic covering map of the Riemann surface, then the Poincaré metric on the unit disc $\Delta$ “pushes down” to a smooth Riemannian (Hermitian) metric on $M$, i.e., there is a unique metric on $M$ such that $\pi$ is a local isometry. This canonical push-down metric on such $M$’s has the property that, if $F\colon M_{1}\to M_{2}$ is a holomorphic map, then $F$ is metric nonincreasing. This property will be exploited early in the paper to recover various classical results on maps of annular regions in $\mathbb{C}$ from the viewpoint of minimization of lengths of curves in free homotopy classes.

Extension of these ideas to higher dimensional complex manifolds and to $k$-homotopy classes, $k>1$, involves new features: The natural metric to consider is of course the “hyperbolic metric” introduced by S. Kobayashi, which defines a natural extension to all dimensions of the canonical metric construction for Riemann surfaces just discussed. But a new aspect arises: the Kobayashi metric on a hyperbolic manifold (in the sense of Kobayashi) need not be a Riemannian metric in dimension $\geq 2$. It is, however, a Finsler metric with an infinitesimal form known as the Kobayashi-Royden metric [23]. But for such an infinitesimal metric, which is only upper semi-continuous and does not satisfy the triangle inequality in general, it is necessary to exercise care in defining $k$-dimensional volume measure to the image of a $k$-sphere in the manifold. We shall consider primarily the details of this matter only for some subsets of $\mathbb{C}^{n}$, $n\geq 2$ (cf. Sections 9 and 10), although in principle greater generality would be possible with other hypotheses. This relevant general idea is $k$-dimensional Hausdorff measures. These ideas are applied in the final sections to obtain results for holomorphic mappings of tubular neighborhoods of totally real $k$-spheres in $\mathbb{C}^{n}$, analogous to the earlier results on maps of annular regions (cf. Section 3). It is also natural to consider the tubular neighborhoods of more general compact totally real submanifolds. For such a general case, the degree of maps, a homology invariant is more appropriate and has been investigated.

• •

  $F^{\textrm{Kob}}_{U}$ : the Kobayashi-Royden metric of an open set $U$ in $\mathbb{C}^{n}$
  

• •

  $\mu^{k}_{X,d}(A)$ : the $k$-dimensional Hausdorff measure of the subset $A$ in the metric space $(X,d)$
  

• •

  $\mu^{k}_{\textrm{Euc}}(B)$ : the $k$-dimensional Hausdorff measure of the subset $B$ in $\mathbb{C}^{n}$ with respect to the Euclidean norm
  

• •

  $\mu^{k}_{\textrm{Kob},U}(C)$ : the $k$-dimensional Hausdorff measure of the subset $C$ of an open set $U$ in $\mathbb{C}^{n}$ with respect to the Kobayashi distance

Let $(X,\rho)$ be a metric space with the metric $\rho$. For a continuous curve $\alpha\colon[a,b]\to X$, and a partition of the interval $[a,b]$

$$P:=\{t_{k}\colon k=0,1,\cdots,N,\textrm{ with }a=t_{0}<t_{1}<\cdots<t_{N}=b\}$$

for some positive integer $N$, let

$$s(\alpha,P):=\sum_{k=1}^{N}\rho(\alpha(t_{k-1}),\alpha(t_{k})).$$

If the set $\{s(\alpha,P)\colon P\textrm{ a partition of }[a,b]\}$ is bounded above, we say that $\alpha$ is rectifiable and define the length $L(\alpha)$ of $\alpha$ by

$$L(\alpha)=\sup\{s(\alpha,P)\colon P\textrm{ a partition of }[a,b]\}.$$

As usual, reparametrizations of rectifiable curves are rectifiable and the length is independent of parametrization.

Notice that, if a map $F$ is distance nonincreasing, then the $F$-images of rectifiable curves are rectifiable and the $F$-images have length $\leq$ the length of the original curve, i.e., length $F(\gamma(t))\leq$ length $\gamma(t)$, for all rectifiable curves $\gamma(t)$ in $X_{1}$.

Distance nonincreasing/length nonincreasing maps arise naturally in complex analysis. The classical Schwarz Lemma is equivalent to the fact that a holomorphic function $f$ from the unit disc $\Delta=\{z\in\mathbb{C}\colon|z|<1\}$ into itself with $f(0)=0$ has the property that $d(0,f(z))\leq d(0,z)$, where $d=$ the Poincaré distance. Since the action by holomorphic isometries of $\Delta$ to itself relative to the Poincaré metric are transitive on $\Delta$, what is known as the Schwarz-Pick Lemma follows immediately: If $f\colon\Delta\to\Delta$ is holomorphic, then $d(f(z_{1}),f(z_{2}))\leq d(z_{1},z_{2})$ for all $z_{1},z_{2}\in\Delta$, where $d$ is the Poincaré distance.

This can be extended to Riemann surfaces as follows: If $M$ is a Riemann surface that is holomorphically covered by the unit disc $\Delta$. say $\pi\colon\Delta\to M$ is a holomorphic covering map, then the covering transformation of the covering $\pi$ are holomorphic isometries for the Poincaré metric. It follows that $M$ has a unique Riemannian (indeed Hermitian) metric for which $\pi$ is locally isometric. Let us call this the canonical metric for $M$.

If $M_{1}$ and $M_{2}$ are two such Riemann surfaces, with canonical (Riemannian) metrics $g_{1}$ and $g_{2}$ respectively, and if $F\colon M_{1}\to M_{2}$ is a holomorphic map, then the pull back $F^{*}g_{2}$ to $M_{1}$ of the metric $g_{2}$, is less than or equal to $g_{1}$, i.e.,

$$F^{*}g_{2}|_{p}\leq g_{1}|_{p}$$

for each $p\in M_{1}$, where

$$F^{*}g_{2}|_{p}(v,w)=g_{2}|_{F(p)}(dF_{p}(v),dF_{p}(w)),$$

for all $v,w$ in the real tangent space of $M_{1}$ at $p$, so $dF_{p}(v)$ and $dF_{p}(w)$ are in the real tangent space of $M_{2}$ at $F(p)$.

To see this distance nonincreasing property, note that if $\pi_{1}\colon\Delta\to M_{1}$ and $\pi_{2}\colon\Delta\to M_{2}$ are holomorphic covering maps, then $F\colon M_{1}\to M_{2}$ can be lifted to a holomorphic map $\hat{F}\colon\Delta\to\Delta$ in such a way that the diagram

$$\begin{CD}\Delta @>{\hat{F}}>{}>\Delta\\ @V{\pi_{1}}V{}V@V{}V{\pi_{2}}V\\ M_{1}@>{}>{F}>M_{2}\end{CD}$$

commutes. The map $\hat{F}$ is nonincreasing for the Poincaré metric by the Schwarz-Pick Lemma. (cf. [14])

It follows, since $\pi_{1}$ and $\pi_{2}$ are (local) isometries, in the sense indicated of the inequality on the pull-back metric, that $F\colon M_{1}\to M_{2}$ is length nonincreasing/distance nonincreasing for the canonical metrics of $M_{1}$ and of $M_{2}$, respectively.

In the case that $M=\{z\in\mathbb{C}\colon\frac{1}{\sqrt{R}}<|z|<\sqrt{R}\}$, $R>1$, which is covered by the unit disc $\Delta$, the canonical metric on $M$ is straightforward to compute. The function $F(z)=e^{iz}$ defines a covering map of the “strip” $\{x+iy\colon-\ln R<y<\ln R\}$ onto $M$, while the strip is biholomorphic to the upper half plane via the composition of a linear map, exponentiation, and a linear fractional transformation. Tracing through gives the formula

$$\frac{\big{(}\pi/(2\ln R)^{2}\big{)}}{r^{2}\cos^{2}\big{(}\pi\ln r/(2\ln R)\big{)}}dr^{2}+\frac{\big{(}\pi/(2\ln R)\big{)}^{2}}{\cos^{2}\big{(}\pi\ln r/(2\ln R)\big{)}}d\theta^{2}$$

for the canonical metric on $M$ in $(r,\theta)$ polar coordinates. (Cf. [8], p. 39.)

The free homotopy classes of closed curves in $M$ are characterized by winding number (around $0$) and are nontrivial for all winding numbers except $0$. Since winding number is $\frac{1}{2\pi}$ times the total change in polar angle $\theta$ around the curve, it follows easily that, for any rectifiable curve $\gamma$ in $M$, the length of the curve in the canonical metric is $\geq\pi^{2}/\ln R$. Thus the $\ell_{1}$-invariant of $M$ in its canonical metric is $\geq\pi^{2}/\ln R$ and indeed

(1)

$$\ell_{1}(M)=\frac{\pi^{2}}{\ln R},$$

with the infimum realized by once-around the curve $r=1$ ($\theta$ goes from $0$ to $2\pi$), either clockwise or counterclockwise. (Throughout, we are using the Poincaré metric on the unit disc $\Delta$ to be $4(1-z\bar{z})^{-2}dz\ d\bar{z}$ so that the curvature $\equiv-1$.)

Since $\{z\in\mathbb{C}\colon A<|z|<B\}$ is linearly biholomorphic to $\{z\colon\sqrt{A/B}<|z|<\sqrt{B/A}\}$, the $\ell_{1}$-invariant of $\{z\in\mathbb{C}\colon A<|z|<B\}$ is equal to $\pi^{2}/\ln(B/A)$. The $\ell_{1}$-invariant is preserved by biholomorphic mapping, so the classical result follows:

This result, originally proved by Hadamard (Cf. [17]), is usually proved by non-metric methods, e.g., Schwarz Reflection, [9, 1, 25] et al.

The nonincreasing property of the $\ell_{1}$-invariant gives an extension of this result:

This Theorem 3.2 implies in particular the historical result:

Notice that the condition on $g$ implies that $g$ is injective on homotopy; For instance the $g$-image of the curve $\gamma$ in the preceding proof will be a closed curve in $\{z\colon A_{2}<|z|<B_{2}\}$ that goes around the origin exactly once. Notice also that the “if” part of this theorem is obviously true by a complex linear map. (Cf. [6, 11] for the original proof.)

A conclusion by the same argument holds for multiply connected domains and Riemann surfaces as well.

The only properties of the “canonical metric” on Riemann surfaces covered by the disc that were crucial here were that holomorphic maps were distance nonincreasing and that the canonical metric was locally comparable (in both directions) with any metric derived from local coordinates. In this context, it is clear that the whole viewpoint has an immediate extension to complex manifolds which are “hyperbolic” in the sense introduced by S. Kobayashi. In this section, we consider only complex manifolds (or complex spaces) that are hyperbolic in the sense of Kobayashi, namely, those for which the Kobayashi pseudodistance is an actual distance function. [15, 16].

In this setting, one can define again the $\ell_{1}$-invariant $\ell_{1}(M)$ of a hyperbolic manifold to be the infimum of the lengths of closed curves that are not freely homotopic to 0 (assuming $M$ is not simply connected). Then following the pattern of before one gets the results:

Thus the results discussed in the concrete instances of Riemann surfaces covered by the unit disc and of annular regions in $\mathbb{C}$ in particular, can be extended to far more general settings. The idea of the $\ell_{1}$-invariant via free homotopy classes of closed curves can also be extended to higher dimensional homotopy classes of maps of the $k$-sphere into complex hyperbolic manifolds, and to some extent, into general length spaces. These methods will be explored in subsequent sections.

The previous discussion of annular regions in $\mathbb{C}$ has a straightforward extension to tubular domains in $\mathbb{C}^{n}$, $n>1$. For this, define, for $0<r<1$,

$$T^{n}(r)=\{\vec{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\colon|z_{1}|=1,\|\vec{z}-(z_{1},0,\ldots,0)\|<r\},$$

where $\|\cdot\|$ is the Euclidean norm. Set $A(r_{1},r_{2})=\{z\in\mathbb{C}\colon r_{1}<|z|<r_{2}\}$ where $0<r_{1}<r_{2}$. The projection map $P_{n}(z_{1},\ldots,z_{n})=z_{1}$ takes $T^{n}(r)$ onto $A(1-r,1+r)$ and there is a natural injection $J_{n}(z)=(z,0,\ldots,0)$ mapping $A(1-r,1+r)$ into $T^{n}(r)$. These maps are homotopy equivalences: $J_{n}$ and $P_{n}$ are homotopy inverses to each other. This observation together with Theorem 3.2 gives rise to a comparison result on these tubular domains (meaning tubular neighborhoods of a circle, where the dimensions need not be equal):

It is natural to consider extending the previous introduced ideas about curve lengths in free homotopy classes of closed curves to higher dimensional homotopy classes. In particular, if $(X,d)$ is a metric space (which we assume arc-wise connected for convenience), then consider continuous maps $f\colon S^{k}\to X$, $k>1$, $S^{k}=\{(x_{0},\cdots,x_{k})\in\mathbb{R}^{k+1}\colon x_{0}^{2}+\ldots+x_{k}^{2}=1\}$ and define two such $f_{0},f_{1}\colon S^{k}\to X$ to be free-homotopic if there is a continuous function $H\colon S^{k}\times[0,1]\to X$ such that, for all $p$, $H(p,0)=f_{0}(p)$ and $H(p,1)=f_{1}(p)$. If one has a situation where, for suitably restricted $f$, one can define the measure of (the image of) such $f\colon S^{k}\to X$, then one could imitate the definition of the $\ell_{1}$-invariant corresponding to the $k=1$ case. But certain difficulties arise: One needs an idea of such a measure and one would want to know that each free homotopy class of continuous maps $S^{k}\to X$ would contain at least one map $f$ with the associated measure of the image of $f$ being finite. This would correspond to rectifiable curves and their lengths, and free-homotopy classes of curves, as in earlier sections.

In the case where $(X,d)$ is a Riemannian manifold and $d$ its Riemannian metric distance, the right idea of measure is, for smooth maps $f\colon S^{k}\to X$, the measure of $f|_{A}$, $A\subset S^{k}$, $A=$ the set of points of $S^{k}$ where the differential $df$ has rank $k$, and the measure of $f|_{A}$ is the usual Riemann-metric induced measure on $k$-dimensional submanifolds. This measure is

$$\mu(f)=\int_{S^{k}}|\textsl{volume form}|$$

where the $|\textsl{volume form}|$ on $(v_{1},\ldots,v_{k})$, where $v_{1},\ldots,v_{k}\in T_{p}S^{k}$, is equal to the absolute value of the $k$-dimensional Riemannian volume at $f(p)$ of $f_{*}(v_{1})\wedge\cdots\wedge f_{*}(v_{k})$. It is easy to show (and well-known) that every free homotopy class of continuous maps $S^{k}\to X$ in this case ($X$ is a Riemannian manifold) contains a smooth (not necessarily everywhere nonsingular) map: If $f$ is a continuous map in the free homotopy class then a sufficiently good smooth approximation of $f$ will be in the same free homotopy class (by deformation along minimal geodesics). So every free homotopy class contains finite $k$-measure maps and hence one can define the infimum of measures over finite-measure (smooth, e.g.) maps in the class and also an $\ell_{k}$ invariant (= infimum over all nontrivial free homotopy classes).

In the case of more general metric spaces, it is not in general clear that any useful concept of $k$-dimensional measure exists. But in the case of metrics which are locally equivalent to Riemannian metrics, the familiar general idea of Hausdorff $k$-dimensional measure, can be used.

For a smooth map $f\colon S^{k}\to M$ of the $k$-sphere $S^{k}$ into a Riemannian manifold $M$, this $\mu^{k}_{M}(f(S^{k}))$ gives essentially the same concept of the $k$-dimensional measure as the Riemannian measure already defined. But this new notion of measure has the advantage that smoothness is not involved.

Rather than exploring these matters further in generality, we restrict our attention now to the specific situation of the Kobayashi metric for a bounded connected open set $U$ in $\mathbb{C}^{n}$. The open set $U$ has its Kobayashi metric in the infinitesimal form, the Kobayashi-Royden metric, as it is usually defined by

$$F^{\textrm{Kob}}_{U}(p,v)=\inf\Big{\{}r>0\colon f\in\mathcal{O}(\Delta,U),f(0)=p,df(0)=\frac{v}{r}\Big{\}}$$

where $\Delta=\{z\in\mathbb{C}\colon|z|<1\}$ and where $\mathcal{O}(\Delta,U)$ denotes the set of holomorphic maps from $\Delta$ to $U$. As is well-known [24, 23], the Kobayashi distance from $p$ to $q$ for $p,q\in U$ is the infimum of the length of the $C^{1}$ curves from $p$ to $q$ in $U$ where the length is taken to be the integral $\int_{\gamma}F^{Kob}_{U}(z,dz)$. (The existence of this integral was also shown in [Op. cit.]). Moreover, it was shown in [2] that $F^{Kob}_{U}(p,v)$ is locally comparable to the standard Euclidean norm of $v$ in $\mathbb{C}^{n}$. This leads easily to the fact that for every free homotopy class of maps $S^{k}\to U$, there is a smooth map $f\colon S^{k}\to U$ (as already noted) and the smooth map will have finite Hausdorff $k$-measure relative to the Kobayashi metric, denoted by $\mu^{k}_{\textrm{Kob},U}(f(S^{k}))$. We call such a map $k$-rectifiable (meaning: Hausdorff $k$-measure is finite), and the $k$-measure just constructed will be called, in this paper, the Hausdorff-Kobayashi $k$-measure.

Thus, one can define an invariant

	$\displaystyle\ell_{k}(U)=$	infimum of the Hausdorff-Kobayashi $k$-measures
		over all the Hausdorff-Kobayashi $k$-rectifiable
		representatives of the nontrivial free homotopy
		classes in $U$

The most natural way to find a bounded connected open set in $\mathbb{C}^{n}$ which has a nontrivial $k$-homotopy class is to take a tubular neighborhood of an embedded $k$-sphere. This is particularly interesting from the viewpoint of complex analysis if one takes the $k$-sphere to be totally real as a submanifold. In particular, the unit $k$-sphere in $\mathbb{R}^{k+1}$ is totally real in $\mathbb{C}^{k+1}$. Let $T_{r}$ be the radius $r>0$ tubular neighborhood

	$\displaystyle T_{r}=\{(x_{0},\ldots,x_{k})+\lambda\vec{u}\colon$	$\displaystyle x_{0}^{2}+\ldots+x_{k}^{2}=1,$
		$\displaystyle\vec{u}\in\mathbb{C}^{k+1},~{}\|u\|=1,~{}0\leq\lambda<r\}.$

For small values of $r$, this is a strongly pseudoconvex domain with $C^{\infty}$ boundary (actually $0<r<\frac{1}{2}$ suffices). According to [7], the Kobayashi-Royden metric goes to $+\infty$ near the boundary relative to the Euclidean metric. Note that there is a natural smooth projection, say $P\colon T_{r}\to S^{k}$, defined by $P(z)=$ the closest point to $z$ in the Euclidean sense in the set $S^{k}$. The injection $S_{k}\hookrightarrow T_{r}$ is a homotopy equivalence with $P$ its homotopy inverse. And the identity (injection) map $S_{k}\to T_{r}$ is homotopically nontrivial. There may be other smooth maps, say $\Gamma\colon S^{k}\to T_{r}$, homotopic to the injection which have the associated Hausdorff-Kobayashi $k$-measure assigned to them being less than the measure assigned to the injection. But the infimum will be realized among the family of maps with image lying in some set of the form

$$\{\vec{x}+\lambda\vec{u}\colon\vec{x}\in S_{k},\vec{u}\in\mathbb{C}^{k+1},~{}\|u\|=1,~{}0\leq\lambda\leq r_{1}\}$$

for some $r_{1}<r$: this follows easily from the estimates on the growth of the Kobayashi metric near the boundary of $T_{r}$ compared to the Euclidean metric. This will yield a conclusion similar to Theorem 3.1, once a technical point about the Kobayashi/Royden infinitesimal metric is established. This will be discussed further in the following sections.

To find some analogue in this situation of Theorems 3.1, one needs to have an idea of strict monotonicity of the Kobayashi distance for the domains (open connected subsets) in $\mathbb{C}^{n}$. Specifically, if two domains $\Omega_{1}$ and $\Omega_{2}$ satisfy $\Omega_{1}\subset\Omega_{2}$, then of course their respective Kobayashi-Royden metrics $F^{\textrm{Kob}}_{\Omega_{1}},F^{\textrm{Kob}}_{\Omega_{2}}$ satisfy $F^{\textrm{Kob}}_{\Omega_{2}}(p,v)\leq F^{\textrm{Kob}}_{\Omega_{1}}(p,v)$ for all $p\in\Omega_{1}$ and $v\in\mathbb{C}^{n}$. But we would like to obtain a strict comparison between them.

Then there exists $\delta>0$ such that the Euclidean distance between $\Omega_{1}$ and $\mathbb{C}^{n}\setminus\Omega_{2}$ is at least $\delta$.

Let $p\in\Omega_{1}$. Notice that there exist positive numbers $b$ and $B$ such that

$$b\leq F^{\textrm{Kob}}_{\Omega_{1}}(p,v)\leq B,~{}\forall v\in\mathbb{C}^{n}\textrm{ with }\|v\|=1,$$

where $\|~{}\|$ denotes the standard Euclidean norm. Fix $p\in\Omega_{1}$ and $v\in\mathbb{C}^{n}$ with $\|v\|=1$. Let $\epsilon>0$ be given arbitrarily, and then choose $h\in\mathcal{O}(\Delta,\Omega_{1})$ satisfying $h(0)=p$, $h^{\prime}(0)=v/r$ for some $r>0$ and

$$F^{\textrm{Kob}}_{\Omega_{1}}(p,v)\leq r<F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon.$$

Take

$$\tilde{h}(z)=h(z)+\delta zv.$$

Then

$$\tilde{h}(0)=h(0)=p,~{}\tilde{h}^{\prime}(0)=h^{\prime}(0)+\delta v=\Big{(}\frac{1}{r}+\delta\Big{)}v$$

and

$$\tilde{h}(z)\subset\Omega_{2},\ \forall z\in\Delta,$$

since $\|\tilde{h}(z)-h(z)\|=\|\delta zv\|<\delta$ for every $z\in\Delta$. So $\tilde{h}\in\mathcal{O}(\Delta,\Omega_{2})$.

This implies

	$\displaystyle F^{\textrm{Kob}}_{\Omega_{2}}(p,v)$	$\displaystyle\leq\frac{1}{1/r+\delta}$
		$\displaystyle=\frac{r}{1+r\delta}$
		$\displaystyle\leq\frac{F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon}{1+\delta F^{\textrm{Kob}}_{\Omega_{1}}(p,v)}$
		$\displaystyle\leq\frac{1}{1+\delta b}(F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon).$

Since $\epsilon>0$ is arbitrary, we obtain that

$$F^{\textrm{Kob}}_{\Omega_{2}}(p,v)\leq\frac{1}{1+\delta b}F^{\textrm{Kob}}_{\Omega_{1}}(p,v)$$

for any $v\in\mathbb{C}^{n}$, due to the homogeneity of the Kobayashi-Royden metric.

Note that $b$ depends on the location of $p$. But on a compact set it stays bounded away from zero, and the desired conclusion follows immediately. $\Box$

The restriction in Lemma 8.1 to a compact set $K$ can be removed.

The results of Section 6 and Section 7 together with the growth of the Kobayashi-Royden metric established in [7] can be combined to establish the results about the domains $T_{r}$ defined in Section 6:

First we note that the $\ell_{k}$-invariants are nonzero in this case. As before, we assume that all $r$-values are small enough that $T_{r}$ is strongly pseudoconvex.

Replace $c$ by $\min\{c,1\}$. It follows that $\ell_{k}(T_{r})\geq c^{k}\ L(T_{r})$, where $L(T_{r})$ represents the infimum of the Euclidean Hausdorff $k$-measure of the image of $S^{k}$ in $T_{r}$ not homotopic to a constant. This latter infimum is positive by elementary considerations. $\Box$

The arguments used to prove Theorem 3.2 can be extended in a straightforward way to prove a corresponding result for the domains of $T_{r}$ type:

The analysis of tubular neighborhoods of totally real embeddings of spheres in the previous section can be extended to more general circumstances. But this extension involves what amounts to a shift from homotopy to homology: the role of being homotopically nontrivial is taken over by having degree not equal to $0$.

The first step is to define the relevant concept of degree: Suppose that $M$ is a smooth ($C^{2}$ suffices) compact connected submanifold without boundary of a Euclidean space $\mathbb{R}^{N}$ and let

$$T_{r}=\bigcup_{w\in M}\{v\in\mathbb{R}^{N}\colon\|v-w\|<r\},$$

where $\|~{}\|$ is the usual Euclidean norm. Notice that there exists $R>0$ such that $T_{r}$, for any $r$ with $0<r<R$, there exists a natural projection $\pi\colon T_{r}\to M$ onto $M$, defined by

$$\pi(v)=\inf_{p\in M}\|p-v\|$$

so that $\pi(v)$ is the closest point to $v$ among points of $M$, with respect to the Euclidean distance. If $g\colon M\to T_{r}$ is a continuous map of $M$ into $T_{r}$, then it is natural to define the degree of $g$, denoted by $\deg g$, to be the degree of $\pi\circ g\colon M\to M$. (If $M$ is orientable, this is to be the usual $\mathbb{Z}$-valued degree. If $M$ is nonorientable, we take degree to be $\mathbb{Z}_{2}$-valued.) If $G\colon T_{r_{1}}\to T_{r_{2}}$ is a continuous map, with $r_{1},r_{2}\in(0,R)$, we define

$$\deg G:=\textrm{the degree of }G\circ i\colon M\to T_{r_{2}},$$

where $i\colon M\to T_{r_{1}}$ is an injection. As well known, these concepts of degree are multiplicative: the degree of a composition is equal to the product of the degrees.

With these definitions in sight, the analogue of Theorem 9.2 is the following:

The conditions on $R$ here are such that

• •

  $T_{r}$ has smooth strongly pseudoconvex boundary and

• •

  the projection map $\pi\colon T_{r}\to M$ is well defined (and continuous in particular),

whenever $0<r<R$.

Note that this result is in fact an extension of Theorem 9.2 since, in the case that $M$ is a sphere, $F$ being homotopic to a constant is equivalent to $\deg F=0$. But in general, of course, $\deg F=0$ does not imply that $F$ is homotopic to a constant. The most obvious example may be the map of $S^{p}\times S^{p}$ to itself, $p\geq 1$, identity on the first factor and constant on the second.

Proof of Theorem 10.1 follows the general pattern of the proofs of previous theorems, but requires some preparation.

First, we need

Now we need

The projection $\pi\colon T_{R}\to M$ admits a constant $C>0$ such that $\|\pi(x)-\pi(y)\|\leq C\|x-y\|$ with respect to the standard Euclidean norm. Thus the Euclidean Hausdorff $k$-volume of $\pi\circ g(M)$ is at least as large as that of $M$, bounded away from $0$.

On the other hand, for $0<r<R$, $T_{r}$ is bounded strongly pseudoconvex. So, by [7], the Kobayashi-Royden metric $F^{\textrm{Kob}}_{T_{r}}$ of $T_{r}$ goes to infinity compared to the Euclidean metric as the base point approaches the boundary of $T_{r}$. Hence there is a constant $c>0$ such that $F^{\textrm{Kob}}_{T_{r}}(q,w)\geq c\|w\|$ for any $q\in T_{r},w\in\mathbb{C}^{n}$. Hence we obtain that

$$\mu^{k}_{\textrm{Kob},T_{r}}(g(M))\geq c^{k}\mu^{k}_{Euc}(g(M)).$$

Since

$$\mu^{k}_{Euc}(g(M))\geq C^{-k}\mu^{k}_{Euc}(\pi(g(M)))\geq C^{-k}\mu^{k}_{Euc}(M),$$

it follows that $\inf_{g}\mu_{\textrm{Kob},T_{r}}^{k}(g(M))\geq(c/C)^{k}\mu^{k}_{Euc}(M)>0$, as desired. $\Box$