V. A. Zorich, “Conformality in the sense of Gromov and holomorphy”, Mat. Sb., 213:11 (2022), 25–30; Sb. Math., 213:11 (2022), 1507

In any lecture course on complex analysis lecturers discuss the relationship between the holomorphy of a function $w=f(z)$ and the conformality of the map between two matching domains in the complex plane performed by this function.

The geometric meaning of the modulus and argument of the derivative $f'(z)$ are explained: the modulus of the derivative is the coefficient of isotropic dilation, while its argument is the angle of rotation at the point under consideration.

Not so commonly do lecturers mention that these properties are not independent in the following sense: if a smooth map takes infinitesimal discs to discs, then this map must also preserve the values of angles.

Gromov extended the notion of a conformal mapping to make it applicable to maps between spaces whose dimensions are different. He proposed to say that a map between metric spaces, for example, $F\colon {\mathbb R}^{m} \to {\mathbb R}^{n}$ (${m} \geqslant {n}$), is conformal if at each point it transforms an infinitesimal ball into an infinitesimal ball [1].

For example, any entire holomorphic function $f\colon {\mathbb C}^n \to {\mathbb C} $ defines a conformal mapping in the sense of Gromov.

A trivial but instructive example of a conformal mapping in the sense of Gromov is the orthogonal projection of Euclidean space onto a plane.

In connection with this extension of the concept of conformality Gromov addressed the natural question of which facts in the classical theory can also apply to these mapping.

In particular, is it true that if the map $F\colon {\mathbb R}^{n+1} \to {\mathbb R}^{n}$ is conformal and bounded, then it is constant, provided that $n \geqslant 2$?

In this paper we explain when a map $w=f(z_1, \dots,z_n) $ from a domain in ${\mathbb C}^n$ to the complex plane ${\mathbb C}$ that is conformal in the sense of Gromov is also holomorphic.

§ 2. Statement of the main result

Consider a mapping $w=f(z_1, \dots,z_n)$ from a domain $D_z$ in ${\mathbb C}^n$ to a domain $D_w$ in the complex plane ${\mathbb C}$.

We assume that, viewed as a real map from ${\mathbb R}^{2n}$ to ${\mathbb R}^2$, $f$ is smooth and nonsingular in the following sense: the rank of its differential is maximal (equal to 2) at all points.

Let $w^0$, $z^0$ be a pair of points such that $w^0=f(z^0)$. Here $w^0 \in D_w$ and $z^0=(z^0_1, \dots,z^0_n)$ is a point in the preimage of $w^0$ in $D_z$.

We look at $f$ in a small neighbourhood $O_{z^0}$ of $z^0$. The real rank of $f$ is always equal to 2, so we can assume that the neighbourhood $O_{z^0}$ is foliated by fibres of real dimension $2n - 2$ such that $f$ acts as a projection along these fibres (each fibre projects onto a point).

We call the 2-dimensional real direction that is transversal and orthogonal to such a fibre and intersects it in a point the proper direction of $f$ at this point.

A small platform in the proper direction is mapped homeomorphically to the target plane. However, this also holds for any platform that is transversal to fibres of the projection map. The proper direction is distinguished by the following property: a mapping is conformal in the sense of Gromov if and only if its restriction to the proper direction is locally conformal at the point under consideration. More precisely, the restriction of the differential of the mapping to the proper direction at a point takes discs to discs, so that it is complex linear or complex antilinear.

It is known that not all real 2-dimensional planes are complex lines in ${\mathbb C}^n$. When a proper direction defines a complex line, we say that it is complex.

Theorem 1. Let $w=f(z_1, \dots,z_n)$ be a mapping from a domain $D_z$ in ${\mathbb C}^n$ to a domain $D_w$ in the complex plane ${\mathbb C}$.

If this mapping is holomorphic, then it is conformal in the sense of Gromov.

On the other hand, if it is conformal in the sense of Gromov, then it is holomorphic if and only if its proper directions are complex.

Of course, the map $f$ in this statement must be as described above, that is, smooth and of real rank 2. Note also that conformality in the sense of Gromov does not take account of orientation, so for greater accuracy we must add to the statement that $w=f(z_1, \dots,z_n)$ is either holomorphic or can be made holomorphic by replacing some of the variables by their conjugates.

In the case under consideration, when the mapping $w=f(z)$ acts from the domain $D_z$ in ${\mathbb C}^n$ to the domain $D_w$ in ${\mathbb C}$, the criterion can also be stated as follows: a mapping $w=f(z_1, \dots,z_n)$ which is conformal in the sense of Gromov is holomorphic if and only if the preimages of all points are complex submanifolds.

§ 3. Proof of the main result

3.1

First we verify that if $w=f(z_1, \dots,z_n)$ is holomorphic, then it is conformal in the sense of Gromov. In fact, the nonsingular holomorphic map $w=f(z_1, \dots,z_n)$ acts as a projection along fibres which are complex manifolds of complex dimension ${n-1}$. The proper direction of $f$, which is transversal and orthogonal to these fibres is therefore complex (spans a complex line). The restriction of the differential of $f$ to it is complex linear.

3.2

We turn to the main assertion: a mapping $w=f(z_1, \dots,z_n)$ that is conformal in the sense of Gromov is holomorphic if its proper directions are complex.

Consider a pair of 2-dimensional planes in general position in Euclidean 3-space. Then the orthogonal projection of one of them onto the other transforms round discs in the original plane into ellipses, rather than round discs.

On the other hand, if we consider two planes in complex direction in complex space (that is, a pair of intersecting complex lines), then the orthogonal projection of one plane onto the other transforms a round disc into a round disc again (conformality).¹

We return to the pair of points $w^0$, $z^0$ such that $w^0=f(z^0)$. Since the real rank of $f$ is equal to $2$ everywhere, we can assume, as mentioned above, that a small neighbourhood $O_{z^0}$ of $z^0$ is foliated by fibres of real dimension $2n - 2$, so that $f$ acts as a projection along these fibres (each of which projects onto a point).

Consider the integral 2-dimensional surface $S_{z^0}$ of proper directions of the map in question that passes through $z^0$ in $O_{z^0}$. (Here we assume that such an integral surface exists — it becomes clear below that this provisional assumption is inessential, but it makes the presentation more geometrically transparent).

The mapping $f$ transforms $S_{z^0}$ into a domain $S_{w^0}$ containing $z^0$ in the plane $\mathbb C$.

Since $f$ is conformal in the sense of Gromov, the restriction of $f$ to $S_{z^0}$ takes infinitesimal discs to discs. As $f$ is smooth, we conclude that the map of $S_{z^0}$ onto $S_{w^0}$ is holomorphic or antiholomorphic.

Introducing the conjugate variable $\overline w$ if necessary we may assume that this mapping is holomorphic.

Note that if in place of $S_{z^0}$ we consider a surface $\widetilde S_{z^0}$ that is merely transversal to the fibration along which $f$ acts as a projection, rather than orthogonal to this fibration, then the fact that $f$ is conformal in the sense of Gromov does not mean that the restriction of $f$ to $\widetilde S_{z^0}$ takes infinitesimal round discs to round discs. Generally speaking, in the image we obtain a distribution of ellipses.

However, when $\widetilde S_{z^0}$ is a holomorphic curve in $\mathbb C^n$, the situation is different. Let us look at this case.

By the hypotheses of Theorem 1, at the point $z^0$ the surface $S_{z^0}$ goes in the direction of a complex line. Consider the complex tangents to $\widetilde S_{z^0}$ and $S_{z^0}$ at $z^0$. Then (as recalled above) the orthogonal projection of one complex line (real plane) onto the other maps discs to discs.

Thus, under the assumptions of Theorem 1 the differential of $f$ has the following property: its restriction to any complex plane through $z^0$ is a complex linear map or the conjugate of such a map.

However, $z^0$ is an arbitrary point in the domain of the definition of $f$. Hence the mapping $w=f(z_1, \dots,z_n) $ is either holomorphic or antiholomorphic in each variable.

Replacing the variables that require it by their conjugates we can assume that $f(z_1, \dots,z_n) $ is holomorphic in each variable in its domain of definition,

As is well known, this means that $f(z_1, \dots,z_n)$ is a holomorphic function.

3.3

Now that we have explained all details, we can sum up the above proof as follows.

The restriction of the differential of a conformal mapping in the sense of Gromov to a proper direction is conformal in the classical sense of a mapping between real 2-dimensional objects. If the proper direction is complex, then even the restriction of the differential of this mapping to an arbitrary complex dimension is conformal. In this case the function $w=f(z_1, \dots,z_n) $ is holomorphic (or antiholomorphic) in each of its independent variables. Hence $w=f(z_1, \dots,z_n) $ is holomorphic or it becomes holomorphic after replacing some of its variables by their conjugates.

§ 4. Brief comment

Note that the result of Theorem 1 is local: it concerns the differential of a mapping at a point, so Theorem 1 is clearly also valid when $f$ is defined on an arbitrary complex manifold with Hermitian structure, and not necessarily on ${\mathbb C}^n$.

Of course, the condition that the mapping is smooth can be relaxed in various ways.

For instance, if $w=\varphi (z)$ is a quasiconformal mapping and $\varphi_{ \overline z}=0$ almost everywhere, then $\varphi$ is holomorphic.

Thus we may assume that locally integrable generalized derivatives exist and $\varphi_{ \overline z}=0$ almost everywhere (for instance, see [5]).

The choice of particular assumptions in place of the smoothness of the mapping is understandably dictated by the problem under consideration.

For readers’ information, we recall that in the paper [1], in the footnote at the first page the author provided the address of a website where the much fuller English text of his paper was available. In particular, the reader could find there an extended interpretation of the notions of conformality and quasiconformality and a statement of the question concerning Liouville’s theorem for such mappings, Now this text is available at https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/problems-sept2014-copy.pdf.

§ 5. Concluding observation

We have discussed a mapping $f$ from a domain $D_z \subset {\mathbb C}^n$ to a domain $D_w \subset {\mathbb C}$. The simplest projection $P\colon {\mathbb R}^3 \to {\mathbb R}^2$, which is conformal in the sense of Gromov, is formally not such a map.

However, if we assume that ${\mathbb C}={\mathbb R}^2 \subset {\mathbb R}^3 \subset {\mathbb C}^2$ and the mapping $P$, conformal in the sense of Gromov, is defined on the submanifold ${\mathbb R}^3 \subset {\mathbb C}^2$, then nevertheless the proper direction of $P$, which is orthogonal to the direction of projection, is a complex direction in ${\mathbb C}^2$. Moreover, in this case there clearly exists even an integral surface orthogonal to the fibres of the projection.

For example, consider the mapping $H\colon {\mathbb S}^3 \to {\mathbb S}^2$ induced by the Hopf fibration. Recall that in terms of complex variables $H(z_1, z_2)=z_2/z_1$ for $(z_1, z_2) \in {\mathbb S}^3 \subset {\mathbb C}^2$. This map is conformal in the sense of Gromov. The sphere ${\mathbb S}^3 \subset {\mathbb C}^2$ is foliated by circles which are taken to points in ${\mathbb S}^2=\overline{\mathbb C}$ by $H$. The proper directions of $H$, which are orthogonal to these circles, form the distribution of complex tangents to ${\mathbb S}^3 \subset {\mathbb C}^2$. However, this distribution is nonintegrable (there exists no integral surface orthogonal to the fibres along which $H$ performs the projection).

The mapping $H\colon {\mathbb S}^3 \to {\mathbb S}^2$ can be regarded as the restriction to ${\mathbb S}^3$ of a holomorphic function $F\colon {\mathbb C}^2 \to \overline{\mathbb C}$, where we set $F(z_1, z_2)=z_2/z_1$.

Of course, this restriction turns out to be holomorphic (satisfying the $\mathrm{CR}$-condition) on complex tangents to ${\mathbb S}^3$.

Formally, the restriction of the holomorphic function $F$ to the totally real plane $\Pi \subset {\mathbb C}^2$ defined by the equations $\operatorname{Im} z_1=0$, $\operatorname{Im} z_2=0$ must also be viewed as a holomorphic map $f\colon \Pi \to \overline{\mathbb R} \sim {\mathbb S}^1$: the function $f$ on $\Pi$ also satisfies formally the $\mathrm{CR}$-condition on complex tangents to $\Pi$, because now the set of complex tangents is empty.

These examples suggest the following direction in which Theorem 1 above can be extended. Let $f\colon M \to {\mathbb C}$ be a conformal mapping in the sense of Gromov defined on a submanifold $M$ of ${\mathbb C}^n$. We say that it is holomorphic if it is a $\mathrm{CR}$-function on $M$, where we assume nonetheless that there are some elements of complex structure in (tangent spaces to) $M$.

Taking this approach to the concept of holomorphy of a map defined on a manifold, we obtain the following result.

Theorem 2. Let $f\colon M \to {\mathbb C}$ be a conformal mapping in the sense of Gromov that is defined on a submanifold $M$ of the space ${\mathbb C}^n$.

If its proper directions are complex, then $f$ is a $\mathrm{CR}$-map.

Citation: V. A. Zorich, “Conformality in the sense of Gromov and holomorphy”, Mat. Sb., 213:11 (2022), 25–30; Sb. Math., 213:11 (2022), 1507–1511

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§ 1. Introduction

§ 2. Statement of the main result

§ 3. Proof of the main result

3.1

3.2

3.3

§ 4. Brief comment

§ 5. Concluding observation

Bibliography