Comparison of invariant functions and metrics

It is shown that all invariant metrics and functions on a bounded $${\mathcal{C}^2}$$C2 -smooth domain coincide on an open non-empty subset. The existence of Lempert–Burns–Krantz discs in $${\mathcal{C}^2}$$C2 -smooth domains and other possible applications are also discussed.


Introduction.
The fundamental Lempert Theorem states that c G = l G and γ G = κ G whenever G is convex or a smooth C-convex domain in C n , where c G is the Carathéodory pseudodistance, l G denotes the Lempert function, γ G is the Carathéodory-Reiffen pseudometric, and κ G is the Kobayashi-Royden pseudometric-see definitions in Section 2.
It is well known that if (d G ) is a contractible family of functions (respectively (δ) G a contractible family of pseudometrics), where G goes through the family of all domains in C n , then c G ≤ d G ≤ l G (resp. γ G ≤ δ G ≤ κ G ). Therefore, the Lempert Theorem may be formulated as follows: on any convex or smooth C-convex domain of C n , all invariant metrics are equal.
This result is surprising as the functions and metrics mentioned above are holomorphic objects (and they are holomorphically invariant) and notions of convexity and C-convexity are just algebraic (topological) conditions.
There are some results describing properties of invariant metrics on strongly pseudoconvex domains. For example, c G and k G (k G denotes the Kobayashi pseudodistance, that is the biggest pseudodistance less or equal to l G ) are comparable in the sense that for any ε > 0 there is a compact subset K = K(ε) of a strongly pseudoconvex domain G such that the inequality  It is well known that generally l D does not satisfy a triangle inequality. Therefore, it is natural to consider the so-called Kobayashi (pseudo)distance given by the formula is a family of holomorphically invariant pseudodistances less than or equal to l D }. Clearly, Vol. 102 (2014)

Comparison of invariant functions and metrics 273
The next objects we are dealing with are the Carathéodory (pseudo)distance and the Carathéodory-Reiffen (pseudo)metric Recall that a holomorphic mapping f : Lempert introduced the concept of stationary map. This idea was originally derived by solving the Euler-Lagrange equations for the extremal problem. Let D be a C 2 -smooth, bounded domain. Recall that f : D −→ D is a stationary mapping if (1) f extends to a C 1/2 -smooth mapping on D; ( Note that if D is additionally convex and f is a stationary mapping in D, then Re z − f (ζ), ν D (f (ζ)) < 0 for any z ∈ D and ζ ∈ T. Therefore, for any z ∈ D the equation (z − f (·)) •f (·) = 0 has exactly one solution F (z) in D. One may check that F is a left inverse for f, i.e., F • f = id .

Equality of invariant metrics and functions.
As mentioned in the Introduction, our main result is the following Then there is a non-empty and open subset U of D × D such that Similarly, there is a non-empty and open subset V of D × C n such that Note that the assumption n ≥ 2 is important. Actually, it is well known that where A is an annulus in the complex plane, that is A := {z ∈ C : r < |z| < 1}, where r < 1.
Similarly, the product property of the invariant metrics and pseudodistances ensures that the assumption about the smoothness is important as well (consider the product of annuli).
We would like to point out that Theorem 1 remains true if C 2 -smoothness will be replaced with C 1,1 -smoothness. The differences in the proofs between these two cases are only technical, so for simplicity we shall consider only the C 2 case.

Lemma 1.
Let D ⊂ C n , n ≥ 2, be a domain. Assume that a ∈ ∂D is such that ∂D is C 2 and strongly convex in a neighborhood of a. Then for any neighborhood V 0 of a, there is a non-empty and open U ⊂ (V 0 ∩ D) × (V 0 ∩ D) with the following property: For Proof. Let r be a C 2 defining function in a neighborhood of a. The problem we are dealing with has a local character, and a is a point of strong convexity. Therefore, analyzing the Taylor series, one may simply see that replacing r with r • Ψ, where Ψ is a local biholomorphism near a, we may assume that a = (0, . . . , 0, 1) and a defining function of D near a is of the form r(z) Of course, the similar holds for partial derivatives of h of first order, i.e.
In particular, D α h(0) = 0 for any α ∈ N n 0 such that |α| = 2. Similarly, as in [13] we consider the mappings 1 Note that A t is an automorphism of B n . Let After some elementary calculations, we infer that Take any increasing sequence {t μ } converging to 1.
Observe that it follows from (6) that r tμ | U0 converges to ρ| U0 in the C 2 (U 0 ) topology. Actually, the local uniform convergence of the functions r tμ follows simply from (3). Similarly, making use of (4), one may deduce the local uniform convergence of partial derivatives of the first order. The local uniform convergence of the partial derivatives of r tμ is a consequence of the continuity of the second-order partial derivatives of h.

Comparison of invariant functions and metrics 275
Let χ be a C ∞ smooth function on R such that Since r tμ converges to ρ in the C 2 topology on U 0 , we easily infer thatρ ν converges to ρ in the C 2 topology on C n .
Let D μ be the connected component of {ρ μ < 0} containing 0. Clearly D μ are strongly convex provided that μ is big enough, as ρ ν converges to ρ in the C 2 topology. Since D μ increase to B, we find that the sequence l Dμ decreases l Bn . Since ρ μ converges to ρ in the C 2 topology, we easily deduce that there is a uniform c > 0 such that any geodesic in D μ such that dist(f (0), ∂D ν ) > 1/c is C 1/2 continuous, and its C 1/2 norm depends only on μ providing that μ is sufficiently large. Now we proceed as follows. Let V ⊂ B n × B n be open and such that any geodesic passing through (z, w) ∈ V lies entirely in U 1 = {Re z n > −1/4}. Let W be non-empty, open, and relatively compact in V .
It follows that for any (z, w) ∈ W there is a geodesic f μ in D μ such that f μ (0) = z and f μ (σ μ ) = w for some σ μ > 0. Passing, if necessary, to a subsequence we may assume that f μ converges to a mapping f 0 : D → B n . Since f 0 (0) = z ∈ B n we get that f 0 (D) ⊂ B n . Moreover, the statement of the Lempert Theorem holds on D μ that is c Dν = l Dν , hence we may easily see that f 0 is a complex geodesic in B n passing through (z, w). Then uniqueness, uniform convergence and C 1/2 uniform continuity implies that f μ (D) lies entirely in {Re z n > −1/2} provided that μ = μ(z, w) is big enough-see Lemma 9 below.
Thus, a standard Baire argument implies the existence of an open nonempty subset W of V and, a natural ν 1 such that for any (z, w) ∈ W and ν ≥ ν 1 , a geodesic of D n passing through (z, w) (let us denote it by f ν,(z,w) ) lies entirely in W . Actually, it suffices to apply the Baire theorem to the family Observe that g ν := A ts ν • f ν is a stationary mapping of D. Since g ν maps D onto arbitrarily small neighborhoods of a provided that ν is sufficiently big, we immediately get the assertion.
Proof of Theorem 1. Losing no generality let us assume that 0 ∈ D. Fix a point a in the topological boundary of D whose distance from 0 is the biggest.
Let U ⊂⊂ U be a neighborhood of a with the following property ( †) for any ζ ∈ ∂D ∩ U and any z ∈ D\U , one has the inequality Re z − ζ, ν D (ζ) ≤ 0.
Making use of Lemma 1, we get an open set U in D × D such that for any (z, w) ∈ U there is a weak stationary mapping of D ∩ U passing through (z, w) and entirely contained in D ∩ U . In particular, for any z ∈ U . Since D ∩ U is convex, we find that (9) holds for any z ∈ U . Making use of ( †), we infer that (9) holds on the whole D.
From this we easily deduce that f has a left inverse F : D → D, hence f is a complex geodesic (actually, F (z) may be obtained as a unique solution of the equation z − f (η),f (η) = 0 with unknown η ∈ D, wheref is a dual map of f ).
Let D be a C 2 smooth strongly pseudoconvex domain in C n , and let a ∈ ∂D. The theorem of Fornaess (see [7], Proposition 1) gives a neighborhood B of a, a strictly convex domain C of C n , a mapping Φ : D → C n extending holomorphically to a neighborhood of D such that Φ(D) ⊂ C, Φ(B\D) ⊂ C n \C, Φ −1 (Φ(B)) = B and the restriction Φ| B : B → Φ(B) is biholomorphic (see also [5] where this result is superseded). It follows from the reasoning presented above that we may construct a complex geodesic f in C lying entirely in Φ(B). Then ( Using this standard reasoning, we easily get the following: Remark 2. Note that using the argument similar to the one used in the proof of Theorem 1, one may show that any C 2 -smooth domain admits Lempert-Burns-Krantz discs. In the case when the domain D is C 6 -smooth, Theorem 3 was proved by Lempert in [13] and formulated in the form above in [2]. It should be noted that the Lempert method may be modified so that it works in the case of C 2+ -smooth strictly pseudoconvex domains (see [14] for details). However, it cannot be applied in the C 2 -smooth case (more precisely, the crucial step of Lempert's arguments relied upon the implicit function theorem to the mapping which is not differentiable assuming only C 2 -smoothness).
The proof presented here is just a modification of the argument used in Section 1. Note that we cannot use here a Baire-type argument and more subtle reasoning is necessary (we shall make use of estimates which are postponed to Section 7).
Proof. As mentioned above the proof is just a slight modification of the proof of Theorem 1, so we shall follow it. So we may assume that r(z) = −1+||z|| 2 + h(z − a), where a = (0, . . . , 0, 1) ∈ C n and h is C 2 -smooth in a neighborhood of 0, h(z) = o(||z|| 2 ), as z → 0. Let r t be given by (5) and, similarly as in (7), put where ρ(z) = −1+||z|| 2 , z ∈ C n . First observe that D t := {z ∈ C n : ρ t (z) < 0} is connected, strongly pseudoconvex, and a lies in its boundary provided that t is close enough to 1. Take any open sets U and V such that U is relatively compact in B n , a ∈ V , and any geodesic of B n passing through points z ∈ U and w ∈ V lies entirely in {Re z n > 0}. Let t be big enough (how big enough will be defined later). Fix z ∈ U and take any sequence (a ν ) ⊂ D t converging to a. Since D t is strictly convex, we get that there is a complex geodesic f ν of D t such that f ν (0) = z and f ν (α ν ) = a ν for some α ν ∈ (0, 1) (U ⊂⊂ D t whenever t is close to 1). Clearly α ν tends to 1. Since f ν are uniformly C 1/2continuous, we may find a subsequence of f ν converging to a complex geodesic f z ∈ C 1/2 (D) of D t such that f z (0) = z and f z (1) = a (see also [4], where the similar argument was used). Now, it follows from Lemma 12 that any complex geodesic g of B n for the pair (z, f z (0)) is close (in the sup-norm) to f z . In particular, g(1) ∈ V (if t is big enough). Clearly g is a complex geodesic for z ∈ U and g(1) ∈ V. Thus the image g(D) lies in {Re z n > 0}. Therefore f z lies entirely in {Re z n > −1/2} (provided that t is big enough).
So fixing t sufficiently close to 1, we easily verify thatŨ := A t (U ) may be arbitrarily close to a (as A t (·) → a uniformly on compact subsets of {Re z n > −1}). Since stationary mappings are invariant under biholomorphisms, we infer that g z := A t • f z is an E-mapping in D passing through A t (z) and a, whose image is contained inŨ . Since U may be arbitrary small, we may apply [7]. Since E-mappings are geodesics on convex domains, we easily find that g z is a complex geodesic.

Remark 3.
It is well known that a geodesic f of C 2 -smooth bounded convex domain is C α for any α < 1. Note that if we knew that there is a left inverse for f of class C α , we would be able to formulate the Burns-Krantz theorem for D.
Of course, iff were of C 1 class, wheref is the dual map to f , then making use of the implicit function theorem and the equality f •f = 1, we would easily find a C 1 -smooth left inverse. Note that such a statement was claimed in [16].

Examples.
As mentioned above we have the following Similarly, making use of the product property of invariant metrics, we get that It may be shown (see e.g. [9,10]) that any extremal mapping of D α intersects one of the axes. Therefore, putting D : Proof. We keep the notation from the proof of Theorem 1, we proceed as Lempert in [13]. First recall that with any f geodesic f of D contained entirely in U , we may associate its dual mapf . This means thatf ∈ O(D, C n ) ∩ C 1/2 (D) andf (ζ) = ζp(ζ)ν D (f (ζ)), ζ ∈ T for some positive, C 1/2 -smooth function p.
The equation ( †) means that for any z ∈ D: In particular, for any z ∈ D the equation (z − f (ζ)) •f (ζ) = 0 has exactly one solution ζ ∈ D denoted by F (z). Clearly, F if a left inverse for f . To prove the uniqueness, suppose that f 1 : D → D is a complex geodesic in D passing through (z, w) ∈ U . We have to show that f 1 is equal to f . Since D is bounded, non-tangential limits f * 1 exist almost everywhere on T. Clearly, , ν D (f (ζ)) = 0 for ζ ∈ T. A strong convexity of D in a neighborhood of a implies that f * 1 = f a.e. on T, so the identity principle finishes the proof.
The result presented above may be used in proving that a smooth domain is not biholomorphic (or there is no proper holomorphic mapping) with a domain whose geodesics are not uniquely determined. Below we present such an application.
Then the set of points (z, w) in D × D possessing a unique complex geodesic has empty interior.
Proof. This is a simple consequence of the product formula c D ((z 1 , z 2 ), (w 1 , w 2 )) = max{c D1 (z 1 , w 1 ), c D2 (z 2 , w 2 )} (see e.g. [9]) and the fact that the sublevels {z ∈ D i : c D (z i , w) = α} have empty interior for any w ∈ D i and α > 0 (which clearly follows from the fact that non-constants holomorphic functions are open).
This gives the following corollary (see also [3,8]): Example 8. Let D be a bounded and smooth. Then it is not biholomorphic with a Cartesian product of non-empty domains.
An interesting application of the above result is that there is no domain fulfilling the assumptions of Theorem 13 in [16].

Technical lemmas.
Recall that a complex geodesic of a strictly pseudoconvex domain is C 1/2 -continuous and speaking very generally its C 1/2 depends only on the curvatures of the topological boundary of a domain, its diameter and the distance between f (0) and ∂D. See [13] for details (the results presented there are formulated for extremals in strictly pseudoconvex domains, but their proofs work for complex geodesics in arbitrary strictly pseudoconvex domains.

Lemma 9.
Assume that D ⊂⊂ B n is a bounded C 2 smooth strongly pseudoconvex domain of C n , 0 ∈ D. Let r ∈ C 2 (B n ) be a defining function of D such that Lr(a, X) ≥ α||X|| 2 , a ∈ ∂D, X ∈ C n for some α > 0. Let (r μ ) ⊂ C 2 (B n ) be a sequence converging to r in a C 2 topology on B n . By D μ we denote the connected component of {z ∈ B n : r μ (z) < 0} containing the origin. Fix a compact subset K of D.
Then every l Dμ -geodesic f such that f (0) ∈ K is C 1/2 continuous, and its C 1/2 may be estimated by a constant independent of μ for μ big enough.
Sketch of the proof of Lemma 9. One may easily show that there is c > 0 such that D and D μ are in D(c) for μ >> 1, where D(c) is the family defined in [14]. Thus it suffices to observe that Propositions 7 and 8 of [14] work for C 2smooth domains when we replace the assumption of being an E-mapping by the assumption of being a geodesic (the proofs given there may be taken over verbatim).