The Carathéodory and Kobayashi/Royden Metrics by Way of Dual Extremal Problems
aa r X i v : . [ m a t h . C V ] M a y The Carath´eodory andKobayashi/Royden Metrics byWay of Dual ExtremalProblems Halsey Royden, Pit-Mann Wong, and Steven G. Krantz November 19, 2018
Abstract:
We study the Carath´eodory and Kobayashi met-rics by way of the method of dual extremal problems in functionalanalysis. Particularly incisive results are obtained for convex do-mains. This paper is based on a preprint that was written by Halsey Royden and Pit-MannWong in the early 1980s. These authors never published this paper, and eventually lostinterest in completing the project. Halsey Royden died in 1993. Some years later, thelast author of the present paper (S. G. Krantz) approached Pit-Mann Wong with the ideaof working together to finish the work and produce a publishable article. Wong readilyagreed, but then he became ill and died in 2011. Now Krantz alone is bringing this work tofruition, including ideas from the original paper and some new ideas as well. We also takethe opportunity to correct a number of errors and misprints in the original manuscript.I am happy to thank Halsey Royden for teaching me much of what I know about theKobayashi/Royden metric. Key Words:
Carath´eodory metric, Kobayashi metric, dual extremal problems, sta-tionary discs MR Classification Numbers: Introduction
Let Ω be a bounded domain in C n and let D be the unit disc in the complexplane C . If a, b ∈ Ω, then define the distance function δ Ω ( a, b ) by δ Ω ( a, b ) ≡ inf { ρ D ( ζ , ζ ) : f ∈ Hol( D, Ω) with f ( ζ ) = a and f ( ζ ) = b } . (0 . ρ D ( ζ , ζ ) denotes the integrated Poincar´e distance on the unit disc D (see [KRA2]). The function δ Ω ( · , · ) does not in general satisfy a triangleinequality (see [LEM] as well as [KRA3]). Thus it is not a metric. The Kobayashi/Royden distance K Ω ( a, b ) is the greatest metric which is smallerthan δ Ω ( a, b ). Namely, K Ω ( a, b ) = inf ( m − X j =0 δ Ω ( a j , a j +1 ) : a = a and a m = b ) . If Ω is bounded and convex, then a result of Lempert [LEM, Theorem 1] as-serts that δ Ω ( a, b ) = K Ω ( a, b ). In the present paper we shall follow Lempert’slead and always assume that Ω is bounded and convex (unless explicitlystated otherwise).A standard normal families argument shows that an extremal map alwaysexists for the distance δ Ω . That is, there is a function f : D → Ω holomorphicwith f ( ζ ) = a , f ( ζ ) = b , and δ Ω ( a, b ) = ρ D ( ζ , ζ ).For a point a ∈ Ω and a tangent vector v at a , we recall that the in-finitesimal Kobayashi metric at a in the direction v is defined to be K Ω ( a ; v ) ≡ inf (cid:26) /λ : there exists a holomorphic mapping f : D → Ω with f (0) = a and f ′ (0) = λ v for some λ > (cid:27) . See [KRA1], [KRA2] for details of this matter. Again, by normal families,an extremal mapping for K Ω ( a ; v ) will always exist.In his seminal paper [LEM], Lempert introduced the concept of stationarymap . This ideas was originally derived simply by solving the Euler-Lagrangeequations for the extremal problem in (0.1). A proper holomorphic map f : D → Ω is said to be stationary if, for almost every ζ ∈ ∂D , there exists anumber p ( ζ ) > ζ p ( ζ ) ν ( f ( ζ )) extends holomorphically to a function e f : D → Ω. Here ν ( f ( ζ )) denotes the unit outward normal to2 Ω at the point f ( ζ ). [In fact we may note that Lempert assumed that ∂ Ωis of class C and that f, e f extend to be of class Λ / up to the boundary.We shall ultimately be able to weaken these hypotheses.]A stationary map f , if it exists, has the property that it is necessarily ex-tremal for the Kobayashi distance K Ω ( a, b ) for any pair of points a, b ∈ f ( D ).It is also extremal for K Ω ( a ; v ) for any point a ∈ f ( D ) and tangent vector v at a . In the paper [LEM], Lempert established the existence and uniquenessof stationary maps together with regularity (i.e., smooth dependence on a and v , for example) for Ω a bounded, strongly convex domain with boundaryof class C .The present paper will establish existence of such maps for any bounded,convex domain without any regularity assumption on the boundary . Weshall also be able to say something about the regularity of these discs. Inthe generality that we treat here, the concept of normal vector ν does notnecessarily make sense (it is ambiguous) and must be replaced by supportinghyperplanes. It is still the case that a boundary point may have many (eveninfinitely many) supporting hyperplanes; thus we shall not generally haveuniqueness of stationary maps. Uniqueness in fact will only be provablewhen the boundary is of class C (see Theorem 3 in Section 2 below). Thisuniqueness result will have the following consequence (see Theorem 12, ofSection 3 below): On any convex domain (here we do not need to assumeboundedness), the infinitesimal Kobayashi metric equals the infinitesimalCarath´eodory metric (also the respective integrated distance functions areequal). This result is somewhat surprising, for in general these metrics arequite distinct.We note that a classical result of Bun Wong [WON] asserts that, if theEisenman-Kobayashi volume form EK Ω (defined by way of maps from the n -ball B n in C n into Ω) and the Eisenman-Carath´eodory volume form EC Ω (defined by way of maps from Ω into B n ) are equal at just one point of Ω thenΩ is biholomorphic to B n (see [EIS] for more on these volume forms). If weconstruct the volume elements K n Ω from the usual Kobayashi metric and C n Ω from the usual Carath´eodory metric, then we have the following inequalities: EC Ω ≤ C n Ω ≤ K n Ω ≤ EK Ω for any domain Ω. Our theorem says that the middle inequality is actually The results below will be formulated in the C k category. The minimal regularityresults are obtained when k = 0.
3n equality on any convex domain Ω.
The idea of a stationary mapping entails both an extremal map f for theKobayashi metric and an associated stationary map e f . Rather than workdirectly with the Kobayashi metric, we instead introduce here another ex-tremal problem from Banach space theory which is more closely affiliatedwith convexity. The problem is a linear one (that is, it is minimizing over aBanach space) and it has the property that extremals for this problem arealso extremal for the Kobayashi metric and vice versa. An additional advan-tage to our new approach is that there is a dual extremal problem which canbe analyzed by way of the Hahn-Banach theorem. The extremal for the dualproblem will correspond rather naturally to e f .Now let Ω be a bounded, convex domain in C n which we will assume with-out loss of generality contains the origin. Let p be the Minkowski functionalof Ω given by p ( z ) = inf { λ > z ∈ λ Ω } (1 . z ∈ C n (see [VAL]). The domain Ω is given by Ω = { z ∈ C n : p ( z ) < } . Now let D = [ ζ ] d [ ζ ] d · · · [ ζ k ] d k be a divisor on the unit disc D in C with total degree d = P j d j ; here each d j is a positive integer. Let { a α,β α ∈ C n : 1 ≤ α ≤ k, ≤ β α ≤ d α − } be a set of vectors in C n .We shall be working with the following space of holomorphic mappings (for k a nonnegative integer): L k = L k ( D , d ) = (cid:8) f : D → C n : f is a holomorphic map which is C k up to the boundary and with f ( β α ) ( ζ α ) = a α,β α for 1 ≤ α ≤ k and 0 ≤ β α ≤ d α − (cid:9) . (1.2)Fix an element f ∈ L k ( D , d ); then any other element in this space is ofthe form f + ϕ , where ϕ is a holomorphic mapping, C k up to the closure,on D which vanishes to order d α at each ζ α . In other words, L k ( D , d ) is anaffine space L k ( D , d ) = f + D H kn ( D ) , (1 . H kn ( D ) is the linear space of n -tuples of functions that are holomorphicon D and C k up to the boundary. We will often find it useful in what follows4o identify an element h = ( h , . . . , h n ) ∈ H kn ( D ) with the n ( k + 1)-tuple (cid:0) ( h (0)1 , h (1)1 , h (2)1 , . . . , h ( k )1 ) , ( h (0)2 , h (1)2 , h (2)2 , . . . , h ( k )2 ) , . . . , ( h (0) n , h (1) n , h (2) n , . . . , h ( k ) n ) (cid:1) . We think of this n ( k + 1)-tuple as an ordered tuple of functions so that f ( j ) ℓ is an antiderivative of f ( j +1) ℓ , 0 ≤ j ≤ k −
1, 1 ≤ ℓ ≤ n . Using the supremumnorm on each entry, we see that the set of such n ( k +1)-tuples forms a Banachspace. We can easily pass back and forth between the two representationsfor an element of H kn ( D ). Of course the two norms are equivalent.For f ∈ L k ( D , d ), we introduce these two important quantities: P ( f ) = sup ζ ∈ D p ( f ( ζ )) and m ( D , d ) = inf f ∈ L P ( f ) . (1 . linear extremal problem is to find f ∈ L k ( D , d ) such that P ( f ) = m ( D , d ). We note that P ( f ) < f ( D ) ⊂ Ω. The followingproposition relates this extremal problem to the corresponding problem forthe Kobayashi metric.
Proposition 1
Let Ω be a bounded, convex domain in C n with C k bound-ary. Then we have (i) A holomorphic mapping f : D → Ω with f ( ζ ) = a and f ( ζ ) = b and with k continuous derivatives up to the boundary is extremal for K Ω ( a, b ) if and only if P ( f ) = 1 and f is extremal for m ( D , d ) , where D = [ ζ ][ ζ ] with data { a, b } . (ii) A holomorphic mapping f : D → Ω with f (0) = a and f ′ (0) = v and with k continuous derivatives up to the boundary is extremal for K Ω ( a ; v ) if and only if P ( f ) = 1 and f is extremal for m ( D , d ) , where D = [0] with data { a, v } . Proof:(i)
Without loss of generality, we may assume that ζ = 0 and ζ = t > m ( D , d ) = P ( f ) = 1 and that f is not extremal for K Ω ( a, b ). Then there is a g : D → Ω holomorphic with g (0) = a and g ( s ) = b for some 0 < s < t . The map h ( ζ ) = g ( sζ /t ) clearlysatisfies h (0) = 0 and h ( t ) = b ; thus h ∈ L k ( D , d ) and h ( D ) = g (( s/t ) D ) ⊂ g ( D ) ⊂ Ω. Therefore P ( h ) < m ( D , d ), which is a clear contradiction.Hence f is extremal for K Ω ( a, b ). 5onversely, suppose that f is extremal for K Ω ( a, b ). Since f ( D ) ⊂ Ω, wehave P ( f ) ≤ P ( f ) < f is not extremal for L k ( D , d ), thenthere is a g ∈ L k ( D , d ) with P ( g ) <
1. We claim that there exists a globalmap F : C → C with (a) F (0) = 0; (b) F ( t ) = b ; (c) F ( D ) ⊂ Ω.Assuming this claim for the moment, we complete the proof as follows.For r > F r ( ζ ) = F ( rζ ) is well defined because F is definedon all of C ; also, for some r >
1, we still have F r ( D ) ⊂ Ω. However, we have F r (0) = 0 and F r ( t/r ) = b with t/r < t , contradicting the extremality of f for K Ω ( a, b ).It remains to prove the claim of the last paragraph but one. Rungeapproximation tells us that, for any ǫ >
0, there is a polynomial h : C → C n with h (0) = g (0) and sup ζ ∈ D | h ( ζ ) − g ( ζ ) | < ǫ . We can also find a polynomialmap ϕ : C → C n so that ϕ (0) = 0 and ϕ ( t ) = (1 , , . . . , ψ : C → C n ζ (cid:18)(cid:2) g ( ζ ) − h ( ζ ) (cid:3) ϕ ( ζ ) , (cid:2) g ( ζ ) − h ( ζ ) (cid:3) ϕ ( ζ ) , . . . , (cid:2) g n ( ζ ) − h n ( ζ ) (cid:3) ϕ n ( ζ ) (cid:19) . Clearly we have sup ζ ∈ D | ψ ( ζ ) | < cǫ with c = sup ζ ∈ D | ϕ ( ζ ) | and the polynomial map F ( ζ ) = h ( ζ ) + ψ ( ζ ) satisfies F (0) = a and F ( t ) = b ;that is to say, F ∈ L k ( D , d ). Moreover,sup ζ ∈ D | F ( ζ ) | ≤ sup ζ ∈ D | h ( ζ ) | + sup ζ ∈ D | ψ ( ζ ) |≤ sup ζ ∈ D | g ( ζ ) | + ǫ + cǫ = P ( g ) + (1 + c ) ǫ< ǫ sufficiently small. Hence F ( D ) ⊂ D . That establishes the claim, andcompletes the proof of part (i) . 6 ii) The proof of part (ii) is analogous to that of part (i) , but we include itfor completeness.If f is not extremal for K Ω ( a ; v ), then there exists a function g : D → Ωholomorphic such that g (0) = a and g ′ (0) = λ v with λ real and λ > h ( ζ ) = g ( λ − ζ ) satisfies h (0) = 0 and h ′ (0) = v , so it lies in L k ( D , d ). However, since λ >
1, we have h ( D ) = g ( λ − D ) ⊂ g ( D ) ⊂ Ω.Hence P ( h ) < m ( D , d ), which is a contradiction.Conversely, if P ( f ) < P ( f ) is not extremal, then there exists a func-tion g ∈ L k ( D , d ) with P ( g ) <
1. As in our earlier argument, we claim thatthere is a polynomial mapping F : C → C n with F (0) = 0, F ′ (0) = v ,and F ( D ) ⊂ Ω. If such an F exists, then, for some λ >
1, the mapping F λ ( ζ ) = F ( λζ ) will satisfy F λ (0) = 0, F ′ λ (0) = λ v , and F λ ( D ) ⊂ Ω, contra-dicting the extremality of f for K Ω ( a ; v ). To construct F , we expand g in apower series about the base point ζ = 0 and set F to be the partial sum ofthe first N terms of that series (for some large N ). For N ≥
2, the map F clearly satisfies F (0) = g (0) and F ′ (0) = g ′ (0) = v and, if N is large enough,we also have P ( F ) <
1, that is to say, F ( D ) ⊂ Ω as desired.This completes the proof of parts (i) and (ii) , and hence the proof ofProposition 1.
We first formulate the dual extremal problem in an abstract setting. Thenwe specialize down to the particular situation that applies to our invariantmetrics.Let X be a complex Banach space with dual X ∗ . A nonnegative, real-valued function P on X is called a Minkowski function for X if it satisfies (2.1) P ( x + y ) ≤ P ( x ) + P ( y ); (2.2) P ( λx ) = λP ( x ) for λ ≥ (2.3) there exists a constant c > c − k x k ≤ P ( x ) ≤ c k x k x ∈ X and k k the Banach space norm on X .We see that a Minkowski function is in effect a norm that is comparable tothe given norm k k on X .Given a Minkowski function P on X , we define on X ∗ the function P ∗ ( u ) = sup x = x ∈ X Re u ( x ) P ( x ) for u ∈ X ∗ . (2 . P ∗ is a Minkowski function on X ∗ with the sameconstant c as that for P .For a complex linear subspace Y of X and a point x not in the closure Y of Y , we define m = inf y ∈ Y P ( x − y ) (2 . M = inf { P ∗ ( u ) : u ∈ Y and Re u ( x ) = 1 } , (2 . Y ≡ { u ∈ X ∗ : u ( y ) = 0 for all y ∈ Y } is the annihilator of Y . Since x Y , it is clear that m > linear extremal problem is to find a point x ∈ x + Y so that m = P ( x ). The dual extremal problem is to find a point u ∈ Y with Re u ( x ) = 1so that M = P ∗ ( u ). Our guiding tenet is the following Principle of Duality : Proposition 2
With notation as above, we have (i) mM = 1 ; (ii) there exists a point u ∈ Y with Re u ( x ) = 1 and P ∗ ( u ) = m − , thatis, the dual extremal problem always has a solution; (iii) if x − x ∈ Y and u ∈ Y are such that Re u ( x ) = P ( x ) · P ∗ ( u ) = 1 , then P ( x ) = m and P ∗ ( u ) = M , that is to say, x and u are, respectively,solutions of the extremal and dual extremal problems. Proof:
On the linear span (over the reals R ) of x and Y , we define a reallinear functional f by setting f ( λx + y ) = λ for all λ ∈ R and y ∈ Y . Since P ( λx + y ) = λ ( x + λ − y ) ≥ λm if λ >
0, and since P ( λx + y ) ≥ λ , we conclude that f ( x ) ≤ m − P ( x ) for all x ∈ R x + y . By theHahn-Banach theorem, f can be extended to a real linear functional F on X F ( x ) ≤ m − P ( x ). Now (following Bohnenblust’s original proof) definea complex linear functional u by setting u ( x ) = F ( x ) − iF ( ix ) . Then u ( y ) = F ( y ) − iF ( iy ) = f ( y ) − i ( f iy ) = 0because Y is a complex linear subspace and f annihilates Y by construction.Furthermore, Re u ( x ) = f ( x ) = 1 andRe u ( x ) = f ( x ) ≤ m − P ( x ) ≤ m − c k x k for all x ∈ X . (2 . u we then see that k u k ≤ √ cm − . Thus u is bounded,i.e., u ∈ X ∗ . From the definition of P ∗ and (2.6), we see immediately that M ≤ P ∗ ( u ) ≤ m − . In particular, mM ≤ u ∈ Y with Re u ( x ) = 1, and any y ∈ Y ,we have P ( x − y ) P ∗ ( u ) ≥ Re u ( x − y ) = Re u ( x ) = 1 . Consequently we also have the reverse inequality m · M ≥
1. Hence M = P ∗ ( u ) = m − , completing the proof of (i) and (ii) .For (iii) , we note that x − x ∈ Y and u ∈ Y imply that Re u ( x ) =Re u ( x ), which is equal to 1 by hypothesis. This means that P ∗ ( u ) ≥ M .On the other hand, the inequality mP ∗ ( u ) ≤ P ( x ) P ∗ ( u ) = 1 = mM impliesthat P ∗ ( u ) ≤ M . Thus P ∗ ( u ) = M and P ( x ) = m .Now let us return to the situation of Section 1 where Ω is a bounded,convex domain C n containing 0 and with Minkowski functional p (relative toΩ). We set p ∗ ( w ) = sup z =0 Re [ z · w ] p ( z ) for w ∈ Ω , (2 . z · w ≡ P nj =1 z j w j .To apply the Principle of Duality to this situation, we choose X = C kn ( ∂D ) = space of C k maps from ∂D to C n . This X is a Banach space with norm k f k = X j ≤ k sup ζ ∈ ∂D | f ( j ) ( ζ ) | . (2 . f = ( f , f , . . . , f n ) ∈ X with the n ( k + 1)-tuple (cid:0) ( f (0)1 , f (1)1 , f (2)1 , . . . , f ( k )1 ) , ( f (0)2 , f (1)2 , f (2)2 , . . . , f ( k )2 ) , . . . , ( f (0) n , f (1) n , f (2) n , . . . , f ( k ) n ) (cid:1) . As in our commentary regarding the definition of H kn , we think of this n ( k +1)-tuple as an ordered tuple of functions so that f ( j ) ℓ is an antiderivative of f ( j +1) ℓ ,0 ≤ j ≤ k −
1, 1 ≤ ℓ ≤ n . Using the supremum norm on each entry, we seethat the set of such n ( k + 1)-tuples forms a Banach space. We can easilypass back and forth between the two representations for an element of X . Ofcourse the two norms are equivalent.Also define the Minkowski function P by P ( f ) = sup ζ ∈ ∂D p ( f ( ζ )) . (2 . A kn ( D ) be the subspace of C kn ( ∂D ) consisting of those functions whichextend holomorphically to D . For a divisor D = [ ζ ] d [ ζ ] d · · · [ ζ m ] d m , thespace Y = D A kn ( D ) = (cid:8) ζ d ζ d · · · ζ d m m f : f ∈ A kn ( D ) (cid:9) (2 . X .Any element of the dual space X ∗ is readily seen (by way of the identifi-cation with n ( k + 1)-tuples described above) to extend by the Hahn-Banachtheorem to an element of the n ( k +1)-fold product of M , the space of regular,Borel measures on ∂D . Then a little analysis shows that we may rewrite thefunctional on X as integration against an n -tuple of operators of the form µ + µ (1)1 + µ (2)2 + · · · + µ ( k ) k with all µ p ∈ M . Here parenthetical superscripts denote derivatives. The annihilator of A kn ( D ),which we denote by A kn ( D ) , is (using the representation of elements of A kn ( D )as n ( k + 1)-tuples) the space[ ζ ] H ′ n ( D ) k ≡ (cid:8)(cid:0) ζ ψ ( ζ ) , ζ ψ ( ζ ) , . . . , ζ ψ k ( ζ ) , ζ ψ ( ζ ) , ζ ψ ( ζ ) , . . . , ζ ψ k ( ζ ) , . . . ,ζ ψ n ( ζ ) , ζ ψ n ( ζ ) , . . . , ζ ψ kn ( ζ ) (cid:1) : ψ j : D → C holomorphic such thateach k ψ j k has a harmonic majorant (cid:9) A ( D ) can haveonly Fourier-Stieltjes coefficients with positive index.] See [DUR] and [KRA1]for these ideas. Using integration by parts as above, it is possible to compressthis represenation for an element of the annihilator space into an n -tuple.Hence the annihilator of Y is given by Y = [ ζ ] D − H ′ n ( D ) k . (2 . h ∈ Y ⊂ X ∗ and f ∈ X , the theorem of F. and M. Riesz impliesthat h ( f ) = 12 π Z π f ( e iθ ) · h ( e iθ ) dθ , (2 . f ( ζ ) · h ( ζ ) = P nj =1 f j ( ζ ) h j ( ζ ). Let P ∗ be the Minkowski function (see(2.4)) on X ∗ associated to P (as defined by (2.9)) on X . It is an easy matterto verify that P ∗ ( h ) = 12 π Z π p ∗ ( h ( e iθ )) dθ , (2 . p ∗ is given by (2.7).Now fix an f ∈ L k ( D , d )—see (1.2)—which is continuous on ∂D . By thePrinciple of Duality, there exists an h ∈ Y withRe h ( f ) = 1 and P ∗ ( h ) = M = m − . Let f be an extremal for m = m ( D , d ). Then f has k derivatives whichextend continuously to the boundary, i.e., f ∈ A kn ( D ) ≡ (cid:8) n -tuples of holomorphic functionswith k derivatives on D which extend continuously to the boundary (cid:9) . As a result, f − f vanishes at ζ , ζ so that ( f − f ) h ∈ D Y = [ ζ ] H ′ n ( D ) k .Thus we have 12 π Z π ( f − f ) · h dθ = value at the origin = 011ence h ( f ) = h ( f ). In particular, we obtain that Re h ( f ) = 1. The followingchain of inequalities is now clear (see (2.7), (2.9), (2.13)):1 = 12 π Z π Re [ f ( e iθ ) · h ( e iθ )] dθ ≤ π Z π p ( f ( e iθ )) p ∗ ( h ( e iθ )) dθ ≤ P ( f ) P ∗ ( h ) ≤ P ( f ) m − = 1 . Thus all the inequalities in this last string are actually equalities. We con-clude that P ( f ) = m and from P ( f ) P ∗ ( h ) = 12 π Z p ( f ) p ∗ ( h ) dθ we see that, for almost all θ , we have p ( f ( e iθ )) = m . Analogously, we con-clude also that Re [ f ( e iθ ) · h ( e iθ )] = p ( f ( e iθ )) p ∗ ( h ( e iθ ))almost everywhere. Interpreting these equalities geometrically, we find that,for almost all θ , h ( e iθ ) defines a supporting hyperplane to ∂ Ω α ≡ { z ∈ C n : p ( z ) = α } at the point f ( e iθ ). We summarize these results in the nexttheorem. Theorem 3
Let Ω be a bounded, convex domain in C n with C k boundaryand f ∈ L k ( D , d ) . Then f is extremal for m = m ( D , d ) if and only if thereis a map h ∈ [ ζ ] D − H ′ n ( D ) k such that Re 12 π Z π f ( e iθ ) · h ( e iθ ) dθ = P ( f ) P ∗ ( h ) . Equivalently, f is extremal for m ( D , d ) if and only if there is an h ∈ [ ζ ] D − H ′ n ( D ) k such that, for almost every θ , we have p ( f ( e iθ )) = m and Re f ( e iθ ) · h ( e iθ ) = p ( f ( e iθ )) p ∗ ( h ( e iθ )) . Furthermore, if Ω is strictly convex, then f is unique.[Here strict convexity means that, if z, w ∈ Ω , then tz + (1 − t ) w ∈ Ω for ≤ t ≤ and is in Ω if < t < .] If Ω has boundary which is smooth ofclass C , then h is unique.
12e have already given a proof for one direction of the theorem. Thereverse implication is a consequence of (iii) of Proposition 2. Also, thelast statement of this theorem concerning uniqueness follows from standardresults about supporting hyperplanes of convex domains.
Corollary 4
Let Ω be a bounded, convex domain in C n with C k boundaryand let f ∈ L k ( D , d ) ∩ L k ( D ′ , d ′ ) , where D and D ′ are divisors on D with d ′ = deg D ′ ≥ deg D = d . If f is extremal for L k ( D , d ) , then it is alsoextremal for L k ( D ′ , d ′ ) . Proof:
Since deg D ′ ≥ deg D , there is a meromorphic function ϕ on D whichis a multiple of D ( D ′ ) − and which is positive on ∂D . This assertion can bereduced to considering (combinations of) the following two cases: (1) A simple pole at the origin. In this case ϕ ( ζ ) = 3 + ζ + ζ − is afunction with the desired properties. (2) A simple pole at the origin and a simple zero at ζ = 1 / Thentake ϕ ( ζ ) = 5 / − ( ζ + ζ − ).Now let h be the map in Theorem 3 and let g = ϕ · h . We see that g ∈ ζ ( D ′ ) − H ′ n ( D ) k and, for almost all ζ with | ζ | = 1, we haveRe [ f ( ζ ) · g ( ζ )] = ϕ ( ζ ) · Re [ f ( ζ ) · h ( ζ )] = ϕ ( ζ ) · p ( f ( ζ )) · p ∗ ( h ( ζ )) = p ( f ( ζ )) · p ∗ ( g ( ζ )) . Thus, by Theorem 3, we again conclude that f is extremal for L k ( D ′ , d ′ ).Taking D = [ ζ ][ ζ ] with data { a, b } in Theorem 3 and applying alsoProposition 1 of Section 1, we now have the following. Corollary 5
Let Ω be a bounded, convex domain with C k boundary, andlet f be an extremal map for the Kobayashi distance K Ω ( a, b ) . Then f isalso extremal for L k ( D , d ) with P ( f ) = 1 and there exits h ∈ [ ζ ] D − H ′ n ( D ) k so that, for almost all ζ with | ζ | = 1 , h ( ζ ) is a supporting hyperplane to ∂ Ω at f ( ζ ) . Furthermore, f is unique if Ω is strictly convex and h is also uniqueif Ω has boundary which is smooth of class C . In particular, we see that theextremal discs for the Kobayashi metric extend C k to the boundary of Ω . A consequence of the last corollary is the next result.13 orollary 6 If f is extremal for K Ω ( a, b ) , where Ω is bounded and convex,then f is also extremal for K Ω ( a ′ , b ′ ) and K Ω ( a ′ : v ) for any a ′ , b ′ ∈ f ( D ) andtangent vector v at a ′ . Remark 7
Corollary 6 generalizes a result of Lempert [LEM, Propositions3 and 4] in which he assumed that Ω is strictly convex and f is stationary. Remark 8
Assume that ∂ Ω is smooth of class C and strongly convex, i.e.,there exists a defining function for Ω with positive definite real Hessian. If f is an exteremal mapping for the Kobayashi distance, then f is of class C / on D . For an extremal map we have the following estimates:dist ( f ( ζ ) , ∂ Ω) ≤ C · (1 − | ζ | ) for all ζ ∈ D ; (2 . | f ′ ( ζ ) | ≤ C ′ dist ( f ( ζ ) , ∂ Ω) / − | ζ | for all ζ ∈ D . (2 . f is stationary. However, all one actually needs isthe property that if f is extremal then it is extremal for any two points in itsimage. We have established this latter property (Corollary 6) for extremalmaps.Combining (2.15) and (2.16) we see that | f ′ ( ζ ) | ≤ C ′′ (1 − | ζ | ) / for all ζ ∈ D .
By the noted lemma of Hardy and Littlewood (see [GOL]), this last is equiv-alent to saying that f is C / on D (see [DUR] and [KRA1]).Thus, for a strongly convex domain with C boundary, we may replace“almost everywhere” by “everywhere”in Corollary 5. We also note that,for the proof of (2.15), strong convexity may be replaced by the weakerassumption that there is a constant r > z ∈ ∂ Ω,there is a ball of radius r contained in Ω that is tangent to ∂ Ω at z . As for(2.16), we may also weaken the boundary regularity by the condition thatthere exists a constant R > z ∈ ∂ Ω has a ball of radius R containing Ω and tangent to ∂ Ω at z . The proof of these statements isquite evident from Lempert’s treatment, and we omit the details.14 emark 9 Let Ω be as in Remark 8 and containing the origin. Let f : D → Ω be a holomorphic mapping with f (0) = 0 that is in fact extremal forthe Kobayashi metric K Ω (0 , f ′ (0)). Taking D = [0] and data { , f ′ (0) } inTheorem 1, we obtain h ∈ ζ − H ′ n ( D ) k such thatRe [ f ( e iθ ) · h ( e iθ )] = p ( f ( e iθ )) · p ∗ ( h ( e iθ )) . In other words, h ( ζ ) defines the unique supporting hyperplane at f ( ζ ) for | ζ | = 1. Since ∂ Ω is smooth of class C , the unit outward normal ν to ∂ Ω iswell defined and is related to h ( ζ ) as follows: h ( ζ ) = ϕ ( ζ ) · ν ( f ( ζ )) for all ζ ∈ ∂D (2 . ϕ ( ζ ) >
0. Then the map e f ( ζ ) = ζ h ( ζ ) (2 . D . Also on ∂D the map e f satisfies e f ( ζ ) = ζ ϕ ( ζ ) ν ( f ( ζ )) , (2 . e f in Lempert’s definition of stationary mapping(see [LEM, p. 434]). The fact that e f is of class C / on D can be shown bymodifying the proofs of Propositions 14, 15, 16 in [LEM]. Actually Lempert([LEM, Proposition 5]) showed that, if ∂ Ω has boundary which is smooth ofclass C , then f and e f are of class at least C on D . An easy consequence(see [LEM, Proposition 6]) of this assertion is the following identity: f ′ ( ζ ) · e f ( ζ ) = n X j =1 f ′ j ( ζ ) e f j ( ζ ) ≡ constantfor all ζ ∈ D . By scaling we may assume without loss of generality that thisconstant is 1. Let Ω be a bounded, strongly pseudoconvex domain in C n with boundarythat is smooth of class C . By Remark 9 in Section 2, we know that an15xtremal disc f for the Kobayashi metric on Ω is stationary in the senseof Lempert and, furthermore, f ′ · e f ≡
1. Furthermore, our constructionshows that these extremal discs will be smooth to the boundary. It is thenintuitively clear that one can holomorphically flatten the extremal disc f ( D )by making the supporting hyperplanes at f ( ζ ) along f ( ∂D ) vertical. In factLempert constructed (see [LEM, Proposition 9]) a map Φ : D × C n − → C n with the property that Φ( ζ , , . . . ,
0) = f ( ζ ) for all ζ ∈ D and in point offact Φ is biholomorphic on an open neighborhood of D × { } in D × C n − onto its image. We want to show that Φ is globally biholomorphic. Morespecifically, let Ω ′ ≡ Φ − (Ω) . (3 . Proposition 10
The map Φ (cid:12)(cid:12) Ω ′ : Ω ′ → Ω is biholomorphic. Proof:
First recall the construction of Φ. The relationship f ′ · e f ≡ e f j of e f have no common zeros on D sothat, by an affine change of coordinates on C n , we may assume without lossof generality that e f and e f have no common zeros in D . Thus there existholomorphic functions g and h on D which are of class C up to the boundary ∂D such that g e f + h e f ≡ . (3 . z = Φ ( ζ , . . . , ζ n ) = f ( ζ ) − ζ e f ( ζ ) − g ( ζ ) n X j =3 ζ j e f j ( ζ ) z = Φ ( ζ , . . . , ζ n ) = f ( ζ ) + ζ e f ( ζ ) − h ( ζ ) n X j =3 ζ j e f j ( ζ ) z α = Φ α ( ζ , . . . , ζ n ) = f α ( ζ ) + ζ α for α = 3 , , . . . , n , (3.3)where ( ζ , ζ , . . . , ζ n ) ∈ D × C n − and ( z , z , . . . , z n ) ∈ C n . The map Φ is ofclass C on D × C n − and holomorphic on D × C n − .We want to invert Φ. To this end consider the determinant of the matrix S consisting of coefficients of ζ ′ α ( α ≥
2) in the system of equations (3.3)16efining f ( ζ ) − z :det S ( ζ ) = det f ( ζ ) − z − e f ( ζ ) − g ( ζ ) e f ( ζ ) . . . − g ( ζ ) e f n ( ζ ) f ( ζ ) − z + e f ( ζ ) − h ( ζ ) e f ( ζ ) . . . − h ( ζ ) e f n ( ζ ) f ( ζ ) − z · ·· · ( I n − ) · · f n ( ζ ) − z n , where I n − = identity matrix of dimension ( n − × ( n − . By a direct calculation we find thatdet S ( ζ ) = ( f ( ζ ) − z ) e f ( ζ )+( f ( ζ ) − z ) e f ( ζ )+ n X α =3 ( f α ( ζ ) − z α ) e f α ( g e f + h e f ) . By (3.2) we have, for all ζ ∈ D ,det S ( ζ ) = n X α =1 ( f α ( ζ ) − z α ) e f α ( ζ ) = ( f ( ζ ) − z ) · e f ( ζ ) . (3 . e f (see (2.20)), we have thatRe det S ( ζ ) ζ = Re [( f ( ζ ) − z ) · ϕ ( ζ )] ν ( f ( ζ )) for ζ ∈ ∂D , (3 . ν is the unit outward normal vector to ∂ Ω and ϕ >
0. The hypothesisthat Ω is strongly convex implies that, for ζ ∈ ∂D ,Re det S ( ζ ) ζ > z ∈ Ω < z Ω (3 . ≥ z ∈ ∂ Ω with equality iff z = f ( ζ ) . Now, if z = 0 ∈ Ω, then ζ − det S ( ζ ) = ζ − f · e f is holomorphic on D (recall that we always assume without loss of generality that f (0) = 0) and,17y (3.5) and the argument principle, that det S has a unique zero (of order1) at ζ = 0. If z = 0, then ζ − det S ( ζ ) = ζ − ( f ( ζ ) − z ) · e f ( ζ )has a unique pole of order 1 at the origin and, if z f ( ∂D ), then again by(3.5) and the argument principle, det S has a unique zero in D . Furthermore,this unique zero ζ clearly depends holomorphically on z ∈ Ω (or z Ω, butwe will not need this fact).Now that ζ is uniquely determined by z ∈ Ω, we see from (3.3) that the ζ α s for α ≥ ζ and z . As for ζ , we observed that, since e f and e f do not vanish simultane-ously, it is also uniquely determined (from (3.3)) by z and all the ζ α , α = 2.This completes the proof of Proposition 10. Corollary 11
With the same hypotheses as in Proposition 10, there existsa holomorphic map G : Ω → f ( D ) onto the image of an extremal map f with the property that G (cid:12)(cid:12) f ( D ) : f ( D ) → f ( D ) is the identity map. Proof:
By construction, Φ( D × { } ) = f ( D ), and in fact Φ( ζ , , . . . ,
0) = f ( ζ ). The domain Ω ′ = Φ − (Ω) contains D ×{ } and is contained in D × C n − ,hence the standard projection π from C n onto the first coordinate maps Ω ′ holomorphically onto D × { } . The map G = Φ ◦ π ◦ Φ − clearly satisfies theconditions of the corollary.Recall that the infinitesimal Carath´eodory metric is given by D Ω ( a ; v ) = sup (cid:8) ρ D ( g ( a ) , g ′ ( v )) : g : Ω → D is holomorphic (cid:9) and the corresponding distance function is C Ω ( a, b ) = sup (cid:8) ρ D ( g ( a ) , g ( b )) : g : Ω → D is holomorphic (cid:9) , where ρ D is the Poincar´e metric on D .An immediate consequence of the corollary is that C Ω ( a ; v ) = K Ω ( a ; v ) (3 . a ∈ Ω and v a tangent vector at a . To see this we assume with-out loss of generality that a = f (0) and v = λ − f f ′ (0) for some λ f > f is extremal for K Ω ( a ; v ), that is to say, K Ω ( a ; v ) = λ − f . Now,since f is extremal, we have f ′ · e f ≡ f ′ isnonvanishing. We also know that f is proper (cf. Proposition 1), hence g = f − : f ( D ) → D exists. Composing g with the map G in the corollary,we obtain a holomorphic map from Ω onto D with g ◦ G (cid:12)(cid:12) f ( D ) = f − . Thus C Ω ( a ; v ) ≥ ρ D (cid:0) g ◦ f ( a ); ( g ◦ f ) ′ ( v ) (cid:1) = ρ D (0; λ − f ∂/∂ζ )= λ − f = K Ω ( a ; v ) . Since the reverse inequality is always true for any Ω, we obtain the identity(3.6) for any bounded strongly convex domain with C boundary. This isactually true for any convex domain Ω. For any such domain is exhausted byan increasing union of bounded, strongly convex domains { Ω j } with smoothboundaries of class C ∞ hence, for any a ∈ Ω, we have a ∈ Ω j for j sufficientlylarge. Let f j be extremal for K Ω j ( a ; v ) = 1 /λ j , where f j (0) = a and f ′ j (0) = λ j v .Since Ω j ⊂ Ω j +1 implies that K Ω j ( a ; v ) ≥ K Ω j +1 ( a ; v ), we have that { /λ j } converges to some number c ≥
0. If c = 0, then clearly K Ω ( a ; v ) = 0.If c > K Ω ( a ; v ) < c , then there exists a holomorphic mapping g : D → Ω with g (0) = a , g ′ (0) = λ v , and λ − < c . Thus, for sufficiently small ǫ , themap h ( ζ ) = g ((1 − ǫ ) ζ ) satisfies h ′ (0) = (1 − ǫ ) λv with (1 − ǫ ) λ > /c ≥ λ j .However, for j sufficiently large, h ( D ) ⊂ Ω j and this contradicts theextremality of f j for K Ω j ( a ; v ). Thus K Ω j ( a ; v ) → K Ω ( a ; v ) as j → ∞ . Inan analogous manner one can also show that C Ω j ( a ; v ) → C Ω ( a ; v ). Fromthese assertions it follows that C Ω j ( a ; v ) = K Ω j ( a ; v ) for all j implies that C Ω ( a ; v ) = K Ω ( a ; v ). We have thus completed the proof of the followingtheorem: Theorem 12
Let Ω be a convex domain in C n . Then the Carath´eodorymetric C Ω ( a ; v ) is equal to the Kobayashi metric K Ω ( a ; v ) for any a ∈ Ω andtangent vector v at a . Concluding Remarks
The ideas in [LEM] have proved to be a powerful force in the modern ge-ometric function theory of several complex variables. Lempert’s originalarguments were rather difficult, technical analysis. This paper has beenan attempt to approach some of his ideas with techniques of soft analysis.Along the way, we have been able to weaken some of his hypotheses andobtain stronger results.We hope to explore other avenues of the Lempert theory of extremal discsin future papers. 20 eferences [DUR]
P. L. Duren,
Theory of H p Spaces , Academic Press, 1970. [EIS]
D. Eisenman,
Intrinsic Measures on Complex Manifolds and Holomor-phic Mappings , a Memoir of the American Mathematical Society, Prov-idence, 1970. [GOL]
G. M. Goluzin,
Geometric Theory of Functions of a Complex Variable ,American Mathematical Society, Providence, 1969. [KRA1]
S. G. Krantz,
Function Theory of Several Complex Variables , 2 nd ed.,American Mathematical Society, Providence, RI, 2001. [KRA2] S. G. Krantz,
Complex Analysis: The Geometric Viewpoint , 2 nd ed.,Mathematical Association of America, Washington, D.C., 2004. [KRA3] S. G. Krantz, The Kobayashi Metric, Extremal Discs, and Biholomor-phic Mappings,
Complex Variables and Elliptic Equations , to appear. [LEM]
L. Lempert, La metrique de Kobayashi et la representation des do-maines sur la boule,
Bull. Soc. Math. France [VAL]
F. A. Valentine,
Convex Sets , McGraw-Hill, New York, 1964. [WONG]
B. Wong, Characterization of the unit ball in C n by its automorphismgroup, Inv. Math. [email protected]@math.wustl.edu