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From pseudo-holomorphic functions to the associated real manifold

This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the differential inequalities describing pseudo-holomorphic functions can be used to define a one-real-dimensional manifold (by the vanishing of a function with nonzero gradient), which is here a 1-parameter family of plane curves. On studying the associated envelopes, such a parameter can be eliminated by solving two nonlinear partial differential equations. The classical differential geometry of curves can be therefore exploited to get a novel perspective on the equations describing the global theory of pseudo-holomorphic functions.


I. INTRODUCTION
The progress in complex analysis and differential geometry has led to many important concepts in pure mathematics and mathematical physics from the nineteenth century until recent times. For example, when twistor theory and its applications to general relativity were developed by Penrose [1], and his school, the subject of complex general relativity [2] emerged as a fascinating branch of modern mathematical physics, where the tools of complex differential geometry were applied to find self-dual or anti-self-dual solutions of (vacuum) Einstein equations [3], and also to develop a suitable definition of twistor in curved spacetime [1], without relying on the differential equation that defines Killing spinor fields, but rather considering suitable surfaces (e.g. the totally null α-and β-surfaces [4]).
The very concept of complex manifold involves complex-analytic transition functions. of CP m , the complex torus, the orientable surfaces [5].
From the point of view of complex analysis, the assumption of holomorphic transition functions is nontrivial. By definition, the function is holomorphic if it is a continuous function of the complex variable z, for which the first derivative f ′ (z) exists. This is enough to ensure continuity of f ′ (z) as well [6,7], jointly with the many properties that one learns in introductory courses, including the equivalence with the Weierstrass definition of complex-analytic function of a complex variable, that involves absolute and uniform convergence of a power series in the first place. However, if the assumption of differentiability is no longer made, one can define a one-parameter family of functions of a complex variable, that are holomorphic only if the parameter µ in the definition is set to 1. These are the pseudo-holomorphic functions, beautifully presented by Bers [8], but we here rely upon the Caccioppoli approach [9,10], better suited if one wants to build a global theory, but apparently (much) less known in the literature, maybe because of lack of an English translation. Interestingly, in the theory of pseudoanalytic functions, no use is made of analytic functions' theory, but on the contrary important topics of the latter, e.g. the Picard theorem, appear in a new light, through the analysis of qualitative aspects. In other words, one can obtain elementary proofs of classical theorems, revealing their intimate nature of metric and topological properties. The Cauchy-Riemann differential equations u x = v y , u y = −v x that express the holomorphic nature of a function are then replaced, in the pseudoholomorphic case, by differential inequalities, i.e. simple majorizations replace the equalities among angles of the conformal representation.
Section II defines pseudoholomorphic functions of a complex variable according to Caccioppoli, studying in detail some basic equations in such a definition. Section III presents our definition of pseudoholomorphic function of several (i.e. two or more) complex variables, inspired by the Caccioppoli work in the case of a single complex variable. Since such functions are believed to be more fundamental, one may hope that the manifolds one arrives at by means of them are also more fundamental in a suitable sense. After a review of discwise quasi-conformal maps with n complex variables in Sect. 4, we study in Sect. 5 the 1-parameter family of plane curves associated with a pseudoholomorphic function, while the envelopes for such curves are considered in Sect. 6. Concluding remarks and open problems are presented in Sect. 7, while some technical points are discussed in the appendix.

II. PSEUDOHOLOMORPHIC FUNCTIONS OF A COMPLEX VARIABLE
Let z be the familiar notation for complex variable z = x + iy, x ∈ R, y ∈ R, and let w be the continuous function with image defined in a field A, a bounded open set of the z plane such that all its points are internal points. Let the functions u(x, y), v(x, y) satisfy the following assumptions: (i) u(x, y) and v(x, y) are absolutely continuous in x and y for almost all values of y and x respectively, while their first derivatives u x , u y , v x , v y are square-integrable in every internal portion of A.
is the Jacobian of the map (x, y) → (u, v), Φ(x, y) is the upper limit ϕ(x, y) is the lower limit where ε i tends to zero as △z → 0, for all i = 1, 2, 3, 4. Multiplication and division by (1 − im) on the right-hand side yields therefore The work in Refs. [9,10] uses actually the nomenclature pseudoanalytic, but we prefer to speak of pseudoholomorphic functions, to avoid confusion with the local theory of pseudoanalytic functions, which relies instead upon generalized Cauchy-Riemann equations [11] B ≡ where, according to a standard notation, we have set solved by It is clear from (2.15) that for any generic E, F and G, m + is positive whereas m − is negative. Once we take second derivative of the function A 2 + B 2 with respect to m, after a little algebra, we obtain the simple expression and hence it is clear that one finds from (2.12) that [12] r + = r(m + ) = E + 2F m + + Gm 2 In other words, the maximum and minimum values of r are themselves solutions of the quadratic equation [12] ρ 2 − (E + G)ρ + J 2 = 0, (2.20) and the upper and lower limit (2.3) and (2.4) turn out to obey the identities [9, 10] jointly with the inequalities By virtue of assumption (i), one can build a sequence (u n (x, y), v n (x, y)) of pairs of functions, describing flat surfaces, such that [9] lim n→∞ u n (x, y) = u(x, y), lim uniformly in every closed portion of the field A, and such that their partial derivatives with respect to x and y converge in mean of order 2; u n and v n being functions as smooth as desired, or even polynomials. Hence it follows that on the surface S associated to u(x, y) and v(x, y), areas and lengths have the classical expressions. This means that, if D and L are a domain and a line within A, respectively, the area τ D and the length τ L read as With this nomenclature, τ is the plane transformation of z into w defined by the equations which in turn describe a flat surface S carried by the plane w. Such a map τ is said to be a pseudo-conformal transformation with parameter µ, i.e. a pseudo-conformal representation of S upon A. To every point z 0 of the field A there corresponds a point P (z 0 ) of S, having as trace on the plane w the point w 0 = w(z 0 ).

A. Quasi-conformal maps for real and complex manifolds
For the case of real manifolds we recall, following Ref. [13], that for any pseudo-group of homeomorphisms of Euclidean space one can define the corresponding category of manifolds.
Thus, the full pseudo-group of homeomorphisms, the subgroup of smooth diffeomorphisms, the pseudo-group of quasiconformal maps and the pseudo-group of Lipschitz maps give rise to the theory of topological manifolds, C ∞ manifolds, quasi-conformal manifolds and Lipschitz manifolds, respectively. In particular, a homeomorphism ϕ : The map ϕ is quasiconformal if it is K-quasiconformal for some K ≥ 1. This range of values of K is responsible for distortion of relative distances of nearby points by a bounded factor.
On the other hand, for functions of complex variable, the concept of quasiconformal maps [14] was introduced by Grötzsch, who considered homeomorphisms (2.1) with a positive Jacobian (2.2). Such a map takes infinitesimal circles (cf. the above remarks on distortion of relative distances) into infinitesimal ellipses, and is called quasi-conformal if the eccentricity of these ellipses is uniformly bounded. This condition can be expressed analytically by either of the three equivalent differential inequalities in (2.23) (our µ parameter is the inverse of the Q parameter used by Bers). This property is conformally invariant: if w = w(z) has it, so does the function where F and f are conformal mappings.
Bers [14] calls a function w(z) as in (2.1) quasi-conformal if it is of the form where χ is a quasi-conformal homeomorphism and f is an holomorphic function. This definition is suggested by an important result of Morrey [15] according to which, if w(z) is a quasi-conformal function defined in the unit disk, then it admits the representation is a homeomorphism of the set |z| ≤ 1 onto the set |ζ| ≤ 1 with χ(0) = 0, χ(1) = 1, which satisfies together with its inverse χ −1 a uniform Holder condition, and where f (ζ) is a holomorphic function of the complex variable ζ, |ζ| < 1.
It is a profound result of the work in Ref. [14], which was inspired also by the work of Mori [16], that the geometric definition relying upon Eq. (2.27) is equivalent to the analytic definition given at the beginning of this section, inspired by Morrey, Caccioppoli, Bers and Nirenberg, according to which a continuous function w(z) = u(x, y) + iv(x, y) in a domain D is quasi-conformal if it has L 2 derivatives satisfying the differential inequalities (2.23) almost everywhere. By equivalent we here mean that the geometric and analytic definition imply each other [14]. Note that, in light of the results in Refs. [9,10,14], our pseudoholomorphic functions are also quasi-conformal maps.

III. PSEUDOHOLOMORPHIC FUNCTIONS OF n COMPLEX VARIABLES
Let us proceed now by generalizing the results of previous section to n complex variables.
let w be the continuous function with image (ii) If then J k ≥ 0 almost everywhere in A, and there exists a positive real number µ k ∈]0, 1] such that almost everywhere in A. We then say that the function w is pseudoholomorphic of parameter µ k for all k = 1, 2, ..., n.
Outside of the holomorphic framework, the increment ratio △w △z k has indeed a rich structure as we know already from Sect. 2 because, upon defining one has n increment ratios where (ε i ) k tends to zero as △z k → 0, for all i = 1, 2, 3, 4 and for all k = 1, 2, ....., n.
Multiplication and division by (1 − im k ) on the right-hand side of (3.7) yields therefore and hence With the help of the same calculations leading to Eq. (2.12) we now find where, according to our own notation, we have set To study the maximum or minimum of A 2 k + B 2 k as a function of m k , we have to evaluate its first derivative, which vanishes if m k solves the algebraic equation of second degree solved by It is clear from (3.15) that for any generic E k , F k and G k , (m + ) k is positive whereas (m − ) k is negative. And then once we take second derivative of the function A 2 k + B 2 k with respect to m k , we find, as in Eq. (2.16), , (3.16) and hence it is clear that ∂ 2 In other words, the maximum and minimum values of r k are themselves solutions of the quadratic equation (cf. Eq. (2.20))  (ii) f is K-quasi-conformal (i.e. quasi-conformal with dilatation parameter K = 1 µ ) on each holomorphic plane.
The meaning of condition (ii) is as follows. Once we have a linear map z 1 = a 1 t + b 1 , ..., z n = a n t + b n , (4.1) the a's and b's being complex constants while t is a complex variable, defined on the unit disc {|t| < 1} and whose image lies completely in D, the composite function is always a K-quasi-conformal function in the unit disc {|t| < 1}. The function f is then said to be K-discwise-quasi-conformal, following Ref. [17]. In particular, a 1-discwise-quasiconformal function is a holomorphic function in D. A number of important properties are found to hold, and they are as follows [17].
whose image lies completely in D, the composite function is a K-quasi-conformal function in the unit disc. As a corollary, the concept of K-discwisequasi-conformal function is invariant under holomorphic transformations. This means that, if f (z 1 , ..., z n ) is K-discwise-quasi-conformal in D, and if is a holomorphic transformation from a domain B in (w 1 , ..., w m )-space into D, then the composite function F (w 1 , ..., w m ) = f (ϕ 1 (w 1 , ..., w m ), ..., ϕ n (w 1 , ..., w m )) (4.6) is again K-discwise-quasi-conformal in B. One can therefore define a K-discwise-quasiconformal function as a smooth function which is K-quasi-conformal on every holomorphic surface. This holds not only in a domain D, but also for an arbitrary set, in particular on an analytic subset in the (z 1 , ..., z n )-space.
Another peculiar property is that the sum of two K-discwise-quasi-conformal functions is not always K-discwise-quasi-conformal. For example, each of the functions is K-discwise-quasi-conformal, but their sum, being equal to is not K-discwise-quasi-conformal, because quasi-conformal functions cannot take only real values (having to provide an open mapping) unless they are a constant. with respect to z 1 , ..., z n does not vanish, the real and imaginary parts of a K-discwise-quasiconformal function f = u + iv satisfy the following system of partial differential equations identically: where the notation for the independent variables is the same as in (3.1).
Theorem 4.4. If f is a K-discwise-quasi-conformal function of two complex variables z 1 , z 2 in a domain D, then at every ordinary point (z 0 1 , z 0 2 ) the set given by the equation is a two-dimensional holomorphic surface.
Theorem 4.5. If κ is a given continuous function of 2n real variables x 1 , y 1 , ..., x n , y n in a domain D, and if the modulus of κ is bounded by a constant k 0 < 1, then a solution of class C 1 in D of the system of partial differential equations (4.7) yields a K-discwise-quasiconformal function f = u + iv in D, whose dilatation K is given by The results here recalled are helpful in understanding properties and limits of the quasiconformal pseudoholomorphic framework that we are investigating, and can be compared with the different perspectives considered in Refs. [18][19][20][21][22].

V. THE REAL MANIFOLD ASSOCIATED WITH A PSEUDOHOLOMORPHIC FUNCTION
Since the conditions (2.23) are differential inequalities, it is at first sight problematic to define a pseudoholomorphic manifold, even just in the case of one complex dimension.
However, we may point out that, if ϕ 2 > µJ is fulfilled (we rule out the case of equalities, for which we refer the reader to the Appendix), then also Φ 2 > µJ is fulfilled, because Φ 2 > ϕ 2 .
Moreover, for the same reason, if Φ 2 < J µ holds, a fortiori ϕ 2 < J µ holds as well. We can therefore consider three positive-definite functions α, β and γ depending on (µ; x, y), such that (2.23) is re-expressed in the form Note also that (5.2) leads to µΦ 2 = J − µβ, while (5.1) implies that µϕ 2 = µ 2 J + µα. Thus, on the left-hand side of (5.3), we obtain eventually exact cancellation of terms proportional We can now exploit (2.17) and (2.21) to re-express (5.1) and (5.2) in the form Within this framework, the desired pseudoholomorphic manifold is defined implicitly by the nonlinear equations (5.5) and (5.6) (actually by a single equation equivalent to them, see below), a procedure which is often used in simpler cases [23] in the literature. Upon using (2.2) and (2.13), we here look for u(x, y), v(x, y), α(µ, x, y) and β(µ, x, y) such that (5.5) and which is nothing but Eq. (5.3). In particular, we may look for u(x, y) and v(x, y) in the form u(x, y) = Ax 2 + Bxy + Cy 2 , This ansatz turns Eq. (5.7) into the form where the coefficients are given by For simplicity, we may study first the particular case when (5.10) is fulfilled with (5.14) In the general case, the condition (5.10) is fulfilled if the quadratic form −A µ x 2 −B µ xy−C µ y 2 is positive-definite. For this purpose, we have to study the eigenvalues λ which are the roots of the following equation of second degree: This equation is solved by the two roots (5.17) and λ 1 (resp. λ 2 ) corresponds to the + (resp. −) sign in front of √ δ ABC . Since δ ABC is a sum of squares of real numbers, we are guaranteed that both roots are real. Moreover, (5.18) and hence positivity of λ 2 ensures positivity of λ 1 as well. By virtue of (5.16), positivity of In other words, (5.10) is always fulfilled provided that the following two conditions hold: where A µ , B µ , C µ are displayed in (5.11)-(5.13).
We have still to understand whether our Eq. (5.10) can be seen to define a manifold.
For this purpose, we recall a basic theorem [24], according to which, for a set V of an affine space E to be an hypersurface (i.e. a manifold of dimension N − 1) of class C m of E, it is necessary and sufficient that, ∀a ∈ V , there exists a neighbourhood U(a) of a within E and a scalar function H, defined in U and of class C m , such that, in a reference system, the partial derivatives ∂H ∂x i (a), i = 1, 2, ..., N are not all simultaneously vanishing. In our case, the scalar function pertaining to Eq. (5.10) is the 1-parameter family H(µ; x, y) ≡ A µ x 2 + B µ xy + C µ y 2 + β(µ; x, y) − α(µ; x, y), (5.22) with gradient having components We may therefore obtain a one-dimensional real manifold, i.e. a curve Γ defined implicitly as the set We are therefore looking for a 1-parameter family of curves defined as in (5.25), where the gradient of H has components The desired α and β functions should satisfy α(µ; x, y) < β(µ; x, y), while their gradient should guarantee that the partial derivatives of H in (5.27) and (5.28) are never simultaneously vanishing.

VI. ENVELOPES OF OUR PLANE CURVE
Our real manifold is a plane curve Γ whose equation H(µ; x, y) = 0 (6.1) involves an arbitrary parameter µ ∈ [0, 1[. As is well known, if each of the positions of the curve Γ is tangent to a fixed curve E, the curve E is called the envelope of the curves Γ, which are said to be enveloped by E.
If an envelope E exists, let (x, y) be the point of tangency of E with that one of the curves Γ that corresponds to a certain value µ of the parameter. By definition, the tangents to the curves E and Γ coincide for all values of µ. If δx and δy are two quantities proportional to the direction cosines of the tangent to Γ, and if dx dµ and dy dµ are the derivatives of the unknown functions x = φ(µ) and y = ψ(µ), the necessary condition for tangency is [25]  In other words, the plane curve R(x, y) = 0 consists of two disjoint parts, one of which is the envelope, while the other is the locus of singular points.
When Eq. (6.1) is taken to have the form (5.26), we find that the curve of equation R(x, y) = 0 of the general theory outlined before can be obtained, at least in principle, from Interestingly, the construction of envelopes for the plane curve associated to pseudoholomorphic functions of a complex variable leads to the elimination of the parameter µ ∈ [0, 1[ provided one is able to solve the nonlinear partial differential equations (6.8) and (6.9).
A. The locus of points where H, ∂H ∂µ , ∂ 2 H ∂µ 2 all vanish Let us now consider a particular plane curve H(µ 1 ; x, y) = 0 (6.10) and a nearby curve, resulting from a small change of the parameter, i.e.
H(µ 1 + λ; x, y) = 0. (6.11) This equation can be replaced by which converges, as λ → 0, to the limit equation Inspired by what is done in Ref. [26] in the case of surfaces, we may therefore consider the solution set of the three simultaneous equations H = 0, ∂H ∂µ = 0, H(µ + λ; x, y) = 0, (6.14) the latter of which can be written in the form (6.15) where η = O(λ 3 ). By construction, this scheme is equivalent to studying the equations where the latter equation converges, as λ → 0, to the limit equation In our problem, our Eqs. (6.8) and (6.9) should be supplemented by This scheme has interesting potentialities because, by virtue of (6.9), we can write Eq. (6.18) in the form 2  to our plane curves, the parameter of the general theory can be eliminated by solving two nonlinear partial differential equations. As far as we know, the consideration of such properties never appeared before in the literature, although the basic concepts of our analysis were all well-known, when taken separately.
As far as we can see, no confusion should arise with the construction of pseudoholomorphic submanifolds performed in Ref. [27], where the author starts instead from a compact, connected 4-manifold X with a symplectic form ω, and considers an almost complex structure J for the tangent bundle T X, J being chosen to be compatible with the form ω. Given a compact submanifold Σ of X, Σ is said to be pseudo-holomorphic when J maps the tangent bundle T Σ to itself as a subspace of T X| Σ .
It would be interesting to build one-complex-dimensional manifolds whose transition functions are pseudoholomorphic according to the global theory of our Secs. 2 and 3, but the differential inequalities involved made it difficult for us to make progress along this line. In the present paper we have instead focused on one-real-dimensional manifolds associated to the differential inequalities (2.23) of the global theory. Our next goal will be to get rid of the unknown parameter appearing in the definition of the real manifold by studying the geometry of plane curves, which will be reported in a future work [28]. Now from (A1) one can see that the ratio of the squares of upper and lower limits of the increment ratios Φ 2 which in turn yields for µ 1 < 1 and this minorization is consistent with our definitions.
We note that we can form a nonlinear partial differential equation involving u x 1 , u y 1 , v x 1 , v y 1 by using equations (3.21) and (A1) and writing and Since J 1 = 0, after a little algebra one can find the following differential equation: where we want to solve for u(x 1 , y 1 ) and v(x 1 , y 1 ).
From here onwards we will be writing x 1 = x and y 1 = y and the parameter α = α µ 1 = 1 + 1 µ 1 so that we settle for this partial differential equation Now for µ 1 = 1 i.e. α = 2 we get the class of holomorphic functions with Cauchy-Riemann conditions being satisfied, since in this case one can write (A7) as (u x − v y ) 2 + (u y + v x ) 2 = 0.
In order to get the most general solution for the class of pseudo analytic functions we need to solve for (α − 2)(u x v y − u y v x ) = (u x − v y ) 2 + (u y + v x ) 2 .
As µ 1 = 1 2 and hence α = 3 is an admissible choice we might try to find a solution for the following nonlinear PDE in 2 variables with constant coefficients (u x v y − u y v x ) = (u x − v y ) 2 + (u y + v x ) 2 . (A10) But even before solving this, if we try to saturate the other inequalitiy in (3.23) and work out the case µ 1 = 1 2 we soon run into inconsistencies as we end up getting an equation which is This clearly hints at possible inconsistencies that we head towards with our attempt of saturating the set of inequalities. The conclusion is that we need strict inequalities in order to deal with pseudoholomorphic functions.
Now let us try to deform w(z) by adding to it a small non-holomorphic part ǫz such that the modified functions u(x, y), v(x, y) become u(x, y) = (x 2 − y 2 + ǫx), v(x, y) = (2xy − ǫy).
With this it is easy to see that the Cauchy-Riemann condition breaks down as u x = v y although u y = −v x . This is clearly a non-analytic case, but can we go ahead with this and recover, with a suitable choice of µ < 1, the pseudoholomorphic case?
Let us present a few computational details. The Jacobian defined in (2.2) for this case becomes J(x, y) = 4(x 2 + y 2 ) − ǫ 2 .
However, since ǫ is very small but finite, as (x, y) tends to (0, 0) this condition is violated on a set of finite measure because the Lebesgue measure of the set A ǫ = x, y : 0 ≤ 2(x 2 + y 2 ) < ǫ 2 is nonvanishing. But then the condition for pseudoholomorphicity would not be satisfied almost everywhere. This means that our example fails to provide a pseudoholomorphic function.