THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS 1 ...

THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS ANDREAS FISCHER Abstract. The proof of the Riemann mapping theorem is not constructive. We study versions of it for sets and functions which are definable in an ominimal expansion of the real field. The diffeomorphisms between the subsets and the unit-ball can be chosen definable if we only request them to be continuously differentiable. For many structures expanding the real exponential field we can choose them smooth. For the globally subanalytic structure the diffeomorphisms can be chosen analytic and definable in an o-minimal expansion of it.

1. Introduction The seminal Riemann mapping theorem, one of the central theorems of complex Analysis, states that every proper simply connected open subset U of C is biholomorphic equivalent to the unit-ball. A proof of this theorem can be found in [14]. However, this proof is not constructive. In the present paper, we consider the Riemann mapping theorem for sets and functions which are definable in an o-minimal expansion M of the real field. By definable we always mean definable in M (with parameters in R). For m ∈ N ∪ {∞, ω} and open subsets U, V of R2 , let C m (U, V ) denote the set of m times continuously differentiable and analytic functions, respectively. The purpose of the present paper is to study the following problem: Let U ⊂ R2 be an open definable simply connected set. For which m ∈ N ∪ {∞, ω} is the set U definably C m -diffeomorphic to the unit-ball? This problem was studied by G. Efroymson for the semialgebraic structure in [8]. He showed that every open simply connected semialgebraic set is semialgebraically C ω -diffeomorphic to the unit-ball in R2 . Two main ingredients of his proof are that the analytic closure of a semialgebraic set is semialgebraic and of the same dimension as the original set, and that semialgebraic mappings are piecewise analytic. Both properties do not hold true for arbitrary o-minimal structures, which forces us to consider definable diffeomorphisms of a less regular function class. For arbitrary o-minimal expansions of the real field we prove the following theorem. Theorem 1.1. Let m ∈ N, and let U ⊂ R2 be a definable simply connected open set. Then U is definably C m -diffeomorphic to the unit-ball. 2000 Mathematics Subject Classification. Primary 14P10; Secondary 14P15, 03C64. Key words and phrases. Riemann mapping theorem, o-minimal structures. Research partially supported by NSERC Discovery Grant of Dr. Salma Kuhlmann. 1

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This result can be improved if the exponential function is definable in M, and if every unary definable function is piecewise C ∞ -smooth. Examples of such ominimal structures are the real exponential field Rexp , cf. [20], and its Pfaffian closure, cf. [15]. For these structures we prove the following stronger version of Theorem 1.1. Theorem 1.2. Let M be an o-minimal expansion of Rexp in which every definable unary function is piecewise C ∞ -smooth. Then every open definable simply connected subset of R2 is definably C ∞ -diffeomorphic to the unit-ball. Note that in the previous theorem we do not require C ∞ cell decomposition, see [2, Chapter 7.3] for an introduction to cell decompositions. The structure Ran consisting of all globally subanalytic sets is of special interest. Recently, a new o-minimal structure RQ was generated in [13]. This structure is a proper expansion of RR an , the real field with restricted analytic functions and with all real powers t 7→ tr for t ∈ (0, ∞) and r ∈ R, cf. [16]. In [12] Kaiser proved the following remarkable version of the Riemann mapping theorem. Theorem 1.3 (Kaiser). Let U ( C be an open bounded simply connected Ran definable set. Suppose that the angles at all analytically singular points of the boundary are irrational multiples of π. Then any biholomorphic map from U to the unit-ball is definable in RQ . The sets considered in the previous theorem may not have any cusps, and they must be the open kernel of their closure. We generalize the previous theorem to arbitrary Ran -definable sets by weakening holomorphic to analytic. Theorem 1.4. Let U ⊂ R2 be an Ran -definable simply connected open set. Then U is RQ -definably C ω -diffeomorphic to the unit-ball. This paper is organized as follows. In Section 2 we briefly recall some concepts from o-minimality. In Section 3 we prove Theorem 1.1 and Theorem 1.2. In section 4 we proof Theorem 1.4. 2. Basics and Notation For a set U , the symbols U and ∂U denote the closure and the boundary of U , respectively. We briefly recall the definition of o-minimal structures. For a general introduction to o-minimality we refer the reader to the two excellent surveys about o-minimality, cf. [2] and [4]. The first one is addressed to model theorists and the latter one to geometers. A subset of Rn that is described by a finite set of polynomial inequalities is called semialgebraic. An o-minimal structure M over R is a sequence (Mn )n∈N of sets where each Mn is the collection of definable subsets of Rn such that a) each Mn is a Boolean algebra of sets containing all semialgebraic subsets of Rn , b) finite Cartesian products of definable sets are definable, c) linear projections of definable sets are definable,

THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS

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d) the definable subsets of R have finite boundary. A function f is called definable if its graph Γ(f ) is definable. The collection of all semialgebraic sets forms an o-minimal structure, cf. [1, Chapter 2]. Further examples of o-minimal structures can be found in [5], [6], [7], [12], [16], [18], and [20]. One excellent property of o-minimal structures is their ‘tameness’, which excludes ‘wild’ sets like the graph of sin(1/x) on (0, ∞). This can be seen by considering the dimension of definable sets. The dimension of a non-empty definable set X is the maximal integer dim(X) for which there is a definable subset of X that is definably homeomorphic to Rdim(X) . The dimension of the empty set is −∞. By [2, Theorem 4.1.8], we have the following useful inequality ¡ ¢ dim X \ X < dim(X). In particular, the boundary of an open definable subset U of R2 is of dimension at most 1, and 0-dimensional definable sets are finite. Moreover, the cell decomposition theorem, cf. [2, Chapter 7.3], implies that for every m ∈ N, every definable set of dimension 1 can be partitioned into finitely many C m submanifolds, called cells, whose underlying set is definable and of one of the following forms: a) isolated point, b) I × {a} or {a} × I where I is some open interval and a ∈ R, c) Γ(h) where h : I → R is a definable C m function and I is an open interval. Throughout the paper, we fix an o-minimal structure M over the real field. 3. Proof of Theorem 1.1 Our strategy for proving Theorem 1.1 and Theorem 1.2 is to reduce the problem to semialgebraic sets. 3.1. Reduction to nice sets. First, we construct a finite composition of definable holomorphic diffeomorphisms mapping the original set U to a nice set, which we define now. Definition 3.1. We call a set U ⊂ R2 nice if U is a bounded open simply connected set such that U is the open kernel of its closure, and if there is no point ξ ∈ R2 such that R2 \ (U ∪ {ξ}) is not connected. Lemma 3.2. Let U ⊂ R2 be a definable simply connected open set. Then U is definably biholomorphic equivalent to a nice set. Proof. We generalize the construction of [8, page 74/75] which was performed for semialgebraic sets. Since U is definable, the boundary ∂U can be stratified into finitely many cells. First we consider the cells of dimension 1 which bound U from both sides. Let M be such a cell. Then, because U is simply connected, there exists a definable continuous injective curve C outside of U , containing M , and running to ∞ on one side. Then, by some translation, we may assume that the other end is the origin in R2 , which we interpret as C. The complex function z 7→ z 2 induces a biholomorphic map from a definable open subset of R2 to R2 \ C whose boundary is the preimage of C. The inverse function s breaks up the curve C; in particular, the double-bounding cell M is eliminated. As s is a biholomorphic map, the regularity of the other cells is not worse than before. After finitely many

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steps, the double bounding cells are eliminated. So U is definably biholomorphic equivalent to a definable open simply connected set W which is the open kernel of its closure. Then there might be finitely many points ξ in the boundary of W such that R2 \ (W ∪ {ξ}) has more connected components than R2 \ W . These points are eliminated similarly. ¤ 3.2. Straightening cusps. This subsection is only relevant for Theorem 1.1 and Theorem 1.2. Now we straighten out cusps of nice sets. Let ξ be an element of the boundary of a nice definable set U . Then the inside angle (ξ) of the boundary at ξ is the angle of the half-tangents of ∂U at ξ. The angle is well defined, and (ξ) is a number of [0, 2π]. Definition 3.3. A nice subset U of R2 is called cuspless if (ξ) > 0 for all ξ ∈ ∂U . We eliminate the points of the boundary with vanishing inside angle. This is the first process in which we loose regularity of the diffeomorphisms. If the exponential function is definable, then there is a definable C ∞ function f : (0, ∞) → [0, 1] which equals 1 on (0, 1) and which vanishes in [2, ∞). Let m ∈ N. In the following lemma we prove definable C m straightening and ∞ C straightening of cusps for arbitrary o-minimal expansions of R and o-minimal expansions of Rexp for which definable unary functions are piecewise C ∞ -smooth, respectively. Lemma 3.4. Let U be a definable nice set. a) Let m ∈ N. Then U is definably C m -diffeomorphic to a definable nice cuspless set V . b) If the o-minimal expansion defines the exponential function such that every definable unary function is piecewise C ∞ -smooth, then we may assume that U is definably C ∞ -diffeomorphic to a definable nice cuspless set V . Proof. The proof for both cases is similar. For case (a) let 0 < m ∈ N, and for case (b) let m = ∞. Step 1: Let ξ be a boundary point whose inside angle vanishes. After some translation and rotation, we may assume that ξ = (0, 0), and that the boundary near ξ is given by two definable continuous functions g, h : [0, ε) → R which satisfy g(t) < h(t),

t ∈ (0, ε),

such that g(t) h(t) = 0, and lim = 0. t&0 t t Step 2: We claim that we may assume that h(t) > 0 and g(t) = −h(t) for t being sufficiently small. As the set V is nice, there is a δ > 0 such that

(1)

lim

t&0

(−δ, δ) ∩ U = {(x, y) : 0 < x < δ, g(x) < y < h(x)}. Let ϕ : (0, ∞) → [0, ∞) be a definable C m function which equals 1 in (0, δ/3) and vanishes in [2δ/3, ∞). Define f : (0, ∞) → (0, ∞) by    1 (h(t) + g(t))ϕ(t), if 0 < t < 2δ , 3 f (t) := 2 2δ  0, if t ≥ . 3

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The function f is a definable C m function such that g(t) < f (t) < h(t),

t ∈ (0, δ/3).

In particular, f (t) − g(t) = h(t) − f (t) for 0 < t < δ/3. We define φ : R2 \ {0} → R2 by ( (x, y), if (x, y) 6∈ (−δ, δ)2 , φ(x, y) := (x, y − f (x)), otherwise. By considering φ(U ) in place of U we may assume that h(t) > 0 and g(t) = −h(t) for t ∈ (0, ε), where ε is chosen small enough. Thus the claim is proved. Step 3: By the Monotonicity Theorem, cf. [2, Theorem 3.1.2], we may assume that the function h is strictly increasing and C m -smooth on some interval (0, ε). Let ψ : (0, ∞) → (0, ∞) be the function defined by µ ¶ εt ψ(t) := h . 1 + εt Then ψ is strictly increasing and C m -smooth. Note that 0 < ψ(t) < h(t) for t ∈ (0, ε). Finally, the mapping Φ : R2 \ {0} → R2 \ {0} which is straighting out the cusps is defined by ¡ ¢ (2) Φ(x, y) := x, yψ(x2 + y 2 ) . This function is definable and C m -smooth. Moreover, Φ is injective, because, for fixed x, the function y 7→ yψ(x2 + y 2 ) is strictly increasing and therefore injective. The surjectivity of Φ follows from the fact that the image of y 7→ yψ(x2 + y 2 ) is R. The determinant of the Jacobi matrix of Φ equals ¡ ¢ ¡ ¢ ψ x2 + y 2 + 2y 2 ψ 0 x2 + y 2 which is strictly positive for all (x, y) ∈ R2 \ {0}. Thus, Φ is a definable C m diffeomorphism. Consider Φ−1 : U → Φ−1 (U ). For small positive t, 0 < tψ(2t2 ) < ψ(t) and − ψ(t) < −tψ(2t2 ) so that Φ−1 maps the vanishing inside angle at (0, 0) to a positive angle. As Φ−1 is a C m diffeomorphism from R2 \{0} to R2 \{0}, the number of boundary points with vanishing inside angle of Φ−1 (U ) is less than that of U . So, after finitely many steps, we obtain a definable nice cuspless set V . ¤ A semialgebraic C ω function is called Nash function. Remark 3.5. We do not know whether definable C ω straightening of cusps is possible in every o-minimal structure. For the structure Ran this is true, see Lemma 4.1. There, it is a consequence of the density of the germs of Nash functions in the Hardy field of Ran . This does not apply to every polynomially bounded o-minimal structure with analytic cell decomposition; consider for example RR , the real field with real powers.

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3.3. Reduction to semialgebraic sets. The boundary of a bounded nice cuspless definable set is locally the graph of a Lipschitz continuous function in appropriate linear coordinates. In general, the unary Nash functions are not dense in the definable C 1 functions with respect to the definable C 1 topology. However, for our purposes it is sufficient to have a very weak approximation of definable Lipschitz continuous C 1 functions by Nash functions. Lemma 3.6. Let f : (a, b) → R be a definable Lipschitz continuous C 1 function. Then, for every continuous semilinear function ε : (a, b) → (0, ∞) there is a Lipschitz continuous Nash function g(a, b) → R such that (3)

f (t) − ε(t) < g(t) < f (t),

t ∈ (a, b).

Proof. We may assume that (4)

ε(t) < min{t − a, b − t}. 0

Since f is a bounded function, both limits lim f 0 (t) and lim f 0 (t)

t&a

t%b

exist in R by the Monotonicity Theorem. We denote them by c and d, respectively. Select a ∆ > 0 which is so small that for some δ > 0, ε(t) ε(t) f (t) − < f (a) + (c − ∆)(t − a) < f (t) − , t ∈ (a, a + δ), 3 6 ε(t) ε(t) f (t) − < f (b) + (d − ∆)(t − b) < f (t) − , t ∈ (b − δ, b). 3 6 We apply the Weierstrass Approximation Theorem, cf. [17], to f − ε/3 restricted to [a + δ/2, b − δ/2] and obtain a polynomial p such that ¯ µ ¶¯ ½ · ¸¾ ¯ ¯ ¯p(t) − f (t) − ε(t) ¯ < 1 inf ε(t) : t ∈ a + δ , b − δ . ¯ 3 ¯ 6 2 2 Choose a semialgebraic C 1 partition of unity ϕ1 , ϕ2 , ϕ3 : (a, b) → [0, ∞) subordinate to the sets µ ¶ µ ¶ µ ¶ δ δ δ δ a, a + , a + ,b − , and b − , b , 2 2 2 2 and define h : (a, b) → R by h(t) := ϕ1 (t)(f (a) + (c − ∆)(t − a)) + ϕ2 (t)p(t) + ϕ3 (t)(f (b) + (c − ∆)(t − a)). The function h is continuous, semialgebraic, and satisfies 2 1 f (t) − ε(t) < h(t) < f (t) − ε(t), t ∈ (a, b). 3 6 By [19, Theorem 1] there is a Nash function g : (a, b) → R such that 1 |g(t) − h(t)| < ε(t), t ∈ (a, b). 6 Hence, g satisfies inequality (3). Note that by definability, the function ε is C 1 smooth near a and b. By l’Hospital’s rule ¯ ¯ ¯ ¯¶ ¯ ¯ ¯ µ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ lim g 0 (t)¯ = ¯ lim g(t) − g(a) ¯ ≤ lim ¯ f (t) − f (a) ¯ + ¯ ε(t) ¯ ¯ ¯ ¯ ¯t&a ¯ ¯t&a ¯ t&a t−a t−a t − a¯ which is bounded by inequality (4) and the boundedness of f 0 . Similarly, we see that limt%b |g 0 (t)| is bounded. So g 0 extends continuously to [a, b]. Therefore, g 0 is

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bounded as [a, b] is compact, and therefore, the function g is Lipschitz continuous. ¤ 3.4. Proof of Theorem 1.1. We will use parts of the proof of Theorem 1.1 for the proof of Theorem 1.2. We perform the following proof for C m functions. For this time being, we assume that 0 < m ∈ N. Proof of Theorem 1.1. By the Lemmas 3.2 and 3.4, there is a definable nice subset V of R2 without vanishing inside angles, such that U is definably C m -diffeomorphic to V . By a Lipschitz stratification, cf. [10, Theorem 1.4], of the boundary of V , we may assume that the boundary partitions into finitely many singletons and finitely many sets C1 , . . . , Cn which are graphs of definable Lipschitz continuous C m functions in some suitable linear orthogonal coordinates of R2 . Let h : (a, b) → R be one of these functions, and Γ(h) = C1 , where Γ denotes the graph of a function. The set V is nice. So we may assume that there is a definable continuous function ∆ : (a, b) → (0, ∞) such that {(x, y) : a < x < b, h(x) − ∆(x) < y < h(x)} ⊂ V and {(x, y) : a < x < b, h(x) < y < h(x) + ∆(x)} ∩ V = ∅. Since the boundary of V has no point ξ with inside angle (ξ) = 0, there is even a continuous semilinear function ε : (a, b) → (0, ∞) with ε(t) < (t − a)(b − t) such that S := {(x, y) : a < x < b, h(x) − ε(x) < y < h(x)} ⊂ V. By taking ε/2 in place of ε we may assume that V \S is still nice and has no vanishing inside angles. According to Lemma 3.6, there is a Nash function g : (a, b) → R such that ε(t) h(t) − < g(t) < h(t), t ∈ (a, b). 3 Let p : (−∞, 1] → R be the piecewise polynomial C 1 function ( 0, if t ≤ 0, p(t) := 3 2 −2t + 3t , if 0 < t ≤ 1. We note that p0 (0) = p0 (1) = 0, that p(1) = 1, and that |p0 (t)| < 2 for all t ∈ (−∞, 1). Let φ = (φ1 , φ2 ) : V → R2 be the definable function  if (x, y) 6∈ S, (x, µ y), µ ¶ ¶ (5) φ(x, y) := y − (2g(x) − h(x))  x, y − p (h(x) − g(x)) , if (x, y) ∈ S. 2(h(x) − g(x)) This map is continuously differentiable. The function φ is identity on V \ S. For (x, y) ∈ S, the partial derivative of φ2 with respect to y is µ ¶ φ2 (x, y) 1 y − (2g(x) − h(x)) = 1 − p0 > 0. ∂y 2 2(h(x) − g(x)) Therefore, φ is injective, and the determinant of the Jacobi matrix of φ is always greater than 0. So φ is a definable C 1 diffeomorphism onto its image. Moreover, the image φ(V ) is a nice set without points with vanishing inside angle, and the part of the boundary of V which corresponds to the graph of h is semialgebraic. By applying this procedure to C2 , . . . , Cn , we obtain a semialgebraic simply connected

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subset W of R2 , and a definable C 1 diffeomorphism from U to W . By [9], the sets U and W are definably C m -diffeomorphic. ¤ 3.5. Proof of Theorem 1.2. We fix an o-minimal expansion M of the real exponential field, for which all definable unary functions are piecewise C ∞ -smooth. Proof of Theorem 1.2. We set m = ∞ and follow the proof of Theorem 1.1 until we reach the definition of the function φ in equation (5). Note that g and h are C ∞ -smooth. The functions p is C ∞ -smooth outside of 0. We now exchange the function p by a definable C ∞ function p˜ : (−∞, 1] → [0, 1], which we obtain as follows. Let ρ : R → [0, 1] be a definable C ∞ function which vanishes in (−∞, 0] and equals 1 in [1, ∞). For K > 0 let p˜ : (−∞, 1) be the definable C ∞ function µ ¶ t 3 2 p˜(t) := (−2t + 3t )ρ . K The function p˜ vanishes in (−∞, 0] and satisfies 0 ≤ p˜(t) ≤ 1 for all t ≤ 1 and any choice of K. Note that for t ∈ [0, 1] ¯ ¯ ¯−2t3 + 3t2 ¯ ≤ 1, ¯ ¯ ¯−6t2 + 6t¯ ≤ 3 , 2 and that

µ 0

2

p˜ (t) = (−6t + 6t)ρ

t K

¶

1 + (−2t + 3t ) ρ0 K 3

2

µ

t K

¶ .

The derivative of ρ is a bounded function as well as t 7→ −2t3 + 3t2 on (0, 1). So, by taking K sufficiently large, |˜ p0 (t)| < 2 for t ∈ (−∞, 1]. We define the function φ as in (5) with p˜ in place of p, and continue the proof similar to that of Theorem 1.1 bearing in mind that the function φ is C ∞ -smooth. ¤ Remark 3.7. If M is an o-minimal expansion of Rexp that admits C ∞ cell decomposition, then definable C ∞ manifolds are definably C ∞ -diffeomorphic if they are definably C 1 -diffeomorphic, cf. [11]. So, in this case, Theorem 1.2 is a consequence of Theorem 1.1. Note that the assumptions in Theorem 1.2 are weaker. 4. Proof of Theorem 1.4 In this section, we let M be the globally subanalytic structure Ran . This structure is polynomially bounded, that is, every unary definable function f : R → R is ultimately bounded by a polynomial. 4.1. The structure RQ . For a detailed description of the structure RQ we refer the reader to [12]. The o-minimal structure RQ is a proper expansion of RR an , the structure of restricted analytic functions with real powers. Moreover, RQ admits analytic cell decomposition, and RQ is polynomially bounded, cf. [12, Theorem A and B].

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4.2. Straightening cusps. Lemma 4.1. Let U be a nice Ran -definable set. Then there is an Ran -definable C ω diffeomorphism from U to some nice cuspless Ran -definable set V . Proof. Let ξ be a point of the boundary at which the boundary has a cusp. By translation and rotation, we may assume that the cusp is given by the Ran -definable analytic functions g, h : (0, ε) → R which satisfy h(t) g(t) = lim =0 t&0 t t&0 t

(6)

lim

and h(t) > g(t) for t ∈ (0, ε). Define f := (h+g)/2. Then the function f is analytic. By the Puiseux expansion for Ran -definable unary P functions, cf. [3], there are a δ > 0, an m ∈ N, and a convergent power-series i ai X i such that f (t) =

∞ X

ai ti/m ,

t ∈ (0, δ).

i=0

By equation (6), the first m + 1 coefficients vanish. We now approximate f by a Nash function η as follows. Let 1 ∆(t) := (h(t) − g(t)), t ∈ (0, δ). 4 Note that Ran is polynomially bounded so that ∆(t) > tp for some p ∈ N and t small enough. By selecting ε > 0 sufficiently small, we find an n ∈ N such that ¯ ¯ ¯ ¯ n ∞ ¯ X ¯ ¯ X ¯ ¯ ¯ ¯ i/m i/m ¯ ai t − f (t)¯ = ¯ ai t ¯ < ∆(t) ¯ ¯ ¯ ¯ ¯ i=m+1

i=n+1

for t ∈ (0, ε). Define the function η : (0, ∞) → R by (7)

η(t) :=

n X

ai ti/m .

i=m+1

The function η is a Nash function, such that g(t) < η(t) < h(t) for t ∈ (0, ε). We extend η as semialgebraic function by setting η(t) = 0 for t ≤ 0. Then, by equation (6) and inequality (7), we see that η is a C 1 function. We define a semialgebraic C 1 diffeomorphism Φ : R2 \ {0} → R2 \ {0} by Φ(x, y) = (x, y + η(x)). ˜ : R2 \ {0} → (0, ∞) be a semialgebraic continuous function, such that Let ∆ ˜ 0) < 1 ∆(t), t ∈ (0, ε). ∆(t, 2 According to [19, Theorem 1], there is a Nash diffeomorphism ψ : R2 \{0} → R2 \{0} such that |ψ(x, y) − Φ(x, y)| < ∆(x, y), (x, y) ∈ R2 \ {0}. (8)

By combining the two previous inequalities, we obtain the inequality g(t) +

∆(t) ∆(t) < ψ(t, 0) < h(t) − 2 2

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for small t > 0. By considering ψ −1 (U ) in place of U , we may assume that h(t) > 0 and g(t) < 0 for t ∈ (0, ε). As the structure Ran is polynomially bounded, we find a positive integer N such that tN < min(h(t), −g(t)) for small t > 0. Define the function Φ : R2 → R2 by ¡ ¢ Φ(x, y) := x, y(x2 + y 2 )N ) . Then, similar to the proof of Lemma 3.4 equation (2), the function Φ is a diffeomorphism, and Φ is obviously a Nash function and therefore definable in Ran . After finitely many steps, all vanishing inside angles disappear. ¤ 4.3. Reduction to Theorem 1.3. Lemma 4.2. Let U be a nice cuspless Ran -definable set. Then U is Ran -definably C ω -diffeomorphic to a nice cuspless Ran -definable set V which has no rational angles at non-analytical boundary points. Proof. As U is cuspless, all boundary angles are strictly positive. Let ξ1 , . . . , ξr be non-analytic boundary points of U . Let ξ be one of these points such that (ξ) ∈ {π, 2π}. Let φ be an injective continuous definable curve outside of U whose one end is ξ and which runs to ∞. By applying the function s of the proof of Lemma 3.2, if necessary twice, to ξ, we may assume that (ξ) 6∈ {π, 2π}. As s is holomorphic outside of ξ, no further angle changes. Thus, after finitely many steps, there are no non-analytic boundary points whose angle is in {π, 2π}. Let X := {ξ1 , . . . , ξr } be the set of non-analytic boundary points of U . Choose a coordinate system in which the x-axis is not perpendicular to any half-tangent of ∂U at ξ for every ξ ∈ X. For all c > 0 consider the map φc : R2 → R2 , (x, y) 7→ (x, cy), which is an analytic mapping. For any ξ ∈ X, the slopes a, b of the half tangents at ξ are multiplied by c by the mapping φc , and a 6= b, as (ξ) 6= π. Hence, the map c 7→ (φc (ξ)) is non-constant and definable in (0, 1). So, for each i = 1, . . . , r, there are only countably many c ∈ (0, 1) such that (φc (ξ)) is a rational angle. Thus, for only countably many values of c, not all angles φc (ξi ), i = 1, . . . , r, are irrational. Hence, there is a c ∈ (0, 1) such that φc (U ) has only irrational angles at non-analytic boundary points. ¤ Proof of Theorem 1.4. By Lemma 3.2, Lemma 4.1 and Lemma 4.2, we may assume that U is Ran -definable C ω -diffeomorphic to a definable nice cuspless set V whose inside angles at non-analytic boundary points are not rational. By Theorem 1.3, there is an RQ -definable biholomorphic map from V to B(0, 1). So U is RQ -definably C ω -diffeomorphic to B(0, 1). ¤ Remark 4.3. The construction of [8] cannot be applied to Ran -definable sets, since it makes use of the fact that the analytic closure of a semialgebraic set is semialgebraic and has the same dimension as the original set. The corresponding property is not true in Ran . (Consider the graph of sin(1/t) for t ≥ 1 which is Ran -definable. Then its analytic closure is R2 . Consider the graph of sin restricted to [0, 1]. Then its analytic closure is the graph of sin which is not definable in any o-minimal structure.)

THE RIEMANN MAPPING THEOREM FOR o-MINIMAL FUNCTIONS

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