aa r X i v : . [ m a t h . C V ] S e p ON THE SZEGŐ METRIC
DAVID BARRETT, LINA LEE
Abstract.
We introduce a new biholomorphically invariant metric based on Fef-ferman’s invariant Szegő kernel and investigate the relation of the new metric tothe Bergman and Carathéodory metrics. A key tool is a new absolutely invariantfunction assembled from the Szegő and Bergman kernels. Introduction
In this paper we introduce the Szegő metric, which is defined similarly to theBergman metric using the Szegő kernel instead of the Bergman kernel. The well-known Szegő kernel S ( z, ζ ) is a reproducing kernel for H ( ∂ Ω) (the closure in L ( ∂ Ω) of the set of holomorphic functions that are continuous up to the boundary); thus f ( z ) = Z ∂ Ω S ( z, ζ ) f ( ζ ) dσ E ( ζ ) , ∀ f ∈ H ( ∂ Ω) where σ E stands for the Euclidean surface measure on ∂ Ω . The problem with thisdefinition though is that, unlike the volume measure on Ω , the Euclidean surfacemeasure is not transformed nicely under a biholomorphic mapping. To resolve thisissue, Fefferman introduced the Fefferman surface area measure, σ F (p. 259 of [11]).We define the Szegő metric using the Szegő kernel with respect to the Feffermansurface area measure. Hence it is invariant under biholomorphic mappings.In section 2, we provide background information on the Fefferman surface measureand define the Szegő metric. In section 3, we introduce a biholomorphically invariantfunction SK Ω ( z, w ) which serves to compare the Bergman and Szegő kernels and thenproceed to use this function to derive a number of asymptotic results relating theSzegő and Bergman metrics. In section 4, we show that the Szegő metric is alwaysgreater than or equal to the Carathéodory metric. In section 5 we show that thereis no universal upper bound or positive lower bound for the ratio of the Szegő andBergman metrics. Standing assumption.
We assume throughout this paper that
Ω = { ρ < } ⊂⊂ C n is a strongly pseudoconvex domain with C ∞ boundary. (We note however thatthe Szegő kernel and metric discussed in this paper will be naturally interpretable onmany other domains; transformation laws such as Propositions 1, 2, 3 and Theorem1 below will hold with additional hypotheses on Φ as needed.)
1N THE SZEGŐ METRIC 2 Background
Let H (Ω) be the closure in L ( ∂ Ω) of A (Ω) = O (Ω) ∩ C (Ω) . Then there exists asesqui-holomorphic Szegő kernel S ( z, · ) such that(2.1) f ( z ) = Z ∂ Ω S ( z, ζ ) f ( ζ ) dσ F ( ζ ) , ∀ f ∈ H (Ω) where σ F is the Fefferman measure defined as follows: dσ F ∧ dρ = c n n +1 s − det (cid:18) ρ k ρ j ρ jk (cid:19) ≤ j,k ≤ n dV or equivalently dσ F = c n n +1 s − det (cid:18) ρ k ρ j ρ jk (cid:19) ≤ j,k ≤ n dσ E k dρ k , where σ E is the usual Euclidean surface measure and ρ j = ∂ρ∂z j , ρ jk = ∂ρ∂z j ∂z k .Note that the surface measure σ F does not depend on the choice of the definingfunction ρ ; one can check this letting ˜ ρ = hρ , where h > is a smooth function, andcalculating dσ F with ˜ ρ . Remark . The constant c n used above is a dimensional constant which was leftunspecified in [11] but has been assigned different values later for convenience indifferent contexts: for example, c n = 2 n/ ( n +1) in [1] and c n = 1 in [14]. Proposition 1.
Let
Φ : Ω −→ Ω be a biholomorphic mapping. Then we have Z ∂ Ω | f | dσ ∂ Ω F = Z ∂ Ω | f ◦ Φ | | det J C Φ | nn +1 dσ ∂ Ω F , where σ ∂ Ω j F denotes the Fefferman measure on Ω j for j = 1 , and J C Φ is the complexJacobian matrix of Φ .Proof. Recall that Φ extends to a diffeomorphism between Ω and Ω [9].Let Φ : Ω −→ Ω be a biholomorphic mapping and Ω = { ρ < } . Then we have dσ ∂ Ω F ∧ d ( ρ ◦ Φ) = c n n +1 s − det (cid:18) ρ ◦ Φ) k ( ρ ◦ Φ) j ( ρ ◦ Φ) jk (cid:19) dV Ω . Since (cid:18) ρ ◦ Φ) k ( ρ ◦ Φ) j ( ρ ◦ Φ) jk (cid:19) = (cid:18) J C Φ (cid:19) (cid:18) ρ k ρ j ρ jk (cid:19) (cid:18) J C Φ (cid:19) , we get det (cid:18) ρ ◦ Φ) k ( ρ ◦ Φ) j ( ρ ◦ Φ) jk (cid:19) = det (cid:18) ρ k ρ j ρ jk (cid:19) | det J C Φ | . N THE SZEGŐ METRIC 3
Therefore we have dσ ∂ Ω F ∧ d ( ρ ◦ Φ) = c n | det J C Φ | / ( n +1) · n +1 s − det (cid:18) ρ k ρ j ρ jk (cid:19) (cid:12)(cid:12) det J R Φ − (cid:12)(cid:12) dV Ω = c n | det J C Φ | / ( n +1) · | det J C Φ | − · n +1 s − det (cid:18) ρ k ρ j ρ jk (cid:19) dV Ω = | det J C Φ | − n/ ( n +1) dσ ∂ Ω F ∧ dρ and it follows that dσ ∂ Ω F pulls back to | det J C Φ | nn +1 dσ ∂ Ω F . (cid:3) Proposition 2.
Let
Φ : Ω −→ Ω , Ω , Ω ⊂ C n be a biholomorphic mapping.Assume there exists a well-defined holomorphic branch of (det J C Φ( z )) n/ ( n +1) on Ω .Then we have (2.2) S Ω ( z, w ) = S Ω (Φ( z ) , Φ( w ))(det J C Φ( z )) n/ ( n +1) (cid:16) det J C Φ( w ) (cid:17) n/ ( n +1) , where S Ω j ( z, w ) is the Szegő kernel on Ω j for j = 1 , .Proof. It is obvious that the right hand side of (2.2) is anti-holomorphic with respectto w , so it will suffice to show that it also satisfies the reproducing property.Let f ∈ H (Ω ) . Then we get Z ∂ Ω S Ω (Φ( z ) , Φ( w )) (det J C Φ( z )) n/ ( n +1) (cid:16) det J C Φ( w ) (cid:17) n/ ( n +1) f ( w ) dσ ∂ Ω F ( w )= (det J C Φ( z )) n/ ( n +1) Z ∂ Ω S Ω (Φ( z ) , ˜ w ) (cid:16) det J C Φ(Φ − ( ˜ w ) (cid:17) n/ ( n +1) · f (Φ − ( ˜ w )) | det J C Φ − ( ˜ w ) | n/ ( n +1) dσ ∂ Ω F ( ˜ w ) . Note that (cid:16) det J C Φ(Φ − ( ˜ w ) (cid:17) n/ ( n +1) | det J C Φ − ( ˜ w ) | n/ ( n +1) is holomorphic with re-spect to ˜ w since we have (cid:16) det J C Φ(Φ − ( ˜ w ) (cid:17) n/ ( n +1) | det J C Φ − ( ˜ w ) | n/ ( n +1) = (cid:16) det J C Φ(Φ − ( ˜ w ) (cid:17) n/ ( n +1) | det J C Φ(Φ − ( ˜ w )) | − n/ ( n +1) = (cid:0) det J C Φ(Φ − ( ˜ w )) (cid:1) − n/ ( n +1) . Hence we obtain Z ∂ Ω S Ω (Φ( z ) , Φ( w )) (det J C Φ( z )) n/ ( n +1) (cid:16) det J C Φ( w ) (cid:17) n/ ( n +1) f ( w ) dσ ∂ Ω F ( w )= (det J C Φ( z )) n/ ( n +1) (cid:0) det J C Φ(Φ − (Φ( z ))) (cid:1) − n/ ( n +1) f (cid:0) Φ − (Φ( z )) (cid:1) = f ( z ) as required. (cid:3) N THE SZEGŐ METRIC 4
Definition 1.
We define the Szegő metric on Ω at z in the direction ξ , F Ω S ( z, ξ ) , asfollows: F Ω S ( z, ξ ) = n X j,k =1 ∂ log S Ω ( z, z ) ∂z j ∂z k ξ j ξ k ! / . Remark . Note that one can write S Ω ( z, w ) = P α φ α ( z ) φ α ( w ) where the φ α ’s forman orthnormal basis of H ( ∂ Ω) . Hence S Ω ( z, z ) is a positive strongly plurisubhar-monic function, ensuring that F Ω S ( z, ξ ) is a genuine Kähler metric. The orthonormalexpansion may also be used to show that(2.3) F Ω S ( z, ξ ) ≥ γ Ω | ξ | for some positive constant γ Ω . Remark . Note that F Ω S ( z, ξ ) does not depend on the choice of the dimensionalconstant c n discussed in Remark 1. Proposition 3.
The Szegő metric is invariant under biholomorphic mappings sat-isfying the hypotheses of Proposition 2, i.e, if
Φ : Ω −→ Ω is such a mapping and z ∈ Ω , ξ ∈ T z Ω , then F Ω S ( z, ξ ) = F Ω S (Φ( z ) , J C Φ( z ) ξ ) . Proof.
From (2.2), we have S Ω ( z, z ) = S Ω (Φ( z ) , Φ( z )) | det J C Φ( z ) | n/ ( n +1) . Hence we have log S Ω ( z, z ) = log S Ω (Φ( z ) , Φ( z )) + nn + 1 h log (det J C Φ( z )) + log (cid:16) det J C Φ( z ) (cid:17)i . Let Φ( z ) = w . Then X j,k ∂ log S Ω ( z, z ) ∂z j ∂z k ξ j ξ k = X j,k X l,m ∂ log S Ω (Φ( z ) , Φ( z )) ∂w l ∂w m ∂w l ∂z j ∂w m ∂z k ξ j ξ k = X l,m ∂ log S Ω ( w, w ) ∂w l ∂w m ( J C Φ ( z ) ξ ) l (cid:16) J C Φ ( z ) ξ (cid:17) m . (cid:3) The Szegő metric on the unit ball.
Let B n = { ρ = | z | − < } ⊂ C n .Then det (cid:18) ρ k ρ j ρ jk (cid:19) = − on ∂ B n . Hence dσ ∂ B n F = c n dσ ∂ B n E for S = {| z | = 1 } ⊂ C n and the Szegő kernel for the unitball in C n is given by(2.4) S ( z, ζ ) = 1 c n ( n − π n − z · ζ ) n . One can rewrite (2.4) as follows: Z ∂ B n S ( z, ζ ) f ( ζ ) dσ F ( ζ ) = Z ∂ B n ( n − π n − z · ζ ) n f ( ζ ) dσ E ( ζ ) = f ( z ) , ∀ f ∈ H ( ∂ B n ) . N THE SZEGŐ METRIC 5
If we calculate the Szegő metric for B n at the origin, we get log S ( z, z ) = log (cid:18) ( n − c n · π n (cid:19) − n log(1 − | z | ) , and ∂ log S ( z, z ) ∂z j ∂z k (cid:12)(cid:12)(cid:12) z =0 = n z j z k (1 −| z | ) (cid:12)(cid:12)(cid:12) z =0 = 0 , j = kn | z j | (1 −| z | ) + n −| z | ) (cid:12)(cid:12)(cid:12) z =0 = n, j = k . Hence we have(2.5) F B n S (0 , ξ ) = √ n | ξ | . Remark . Note that the Bergman metric on the unit ball in C n evaluated at theorigin is given as(2.6) F B n B (0 , ξ ) = √ n + 1 | ξ | and the Kobayashi or Carathéodory metric on the unit ball in C n at the origin isgiven as(2.7) F B n K (0 , ξ ) = F B n C (0 , ξ ) = | ξ | . Since all four metrics are invariant under the automorphism group of B n whichacts transitively on B n , relations between the metrics at the origin will propagatethroughout B n . In particular, from (2.5), (2.6) and (2.7) we obtain(2.8) F B n S ( z, ξ ) = √ n F B n C ( z, ξ ) = √ n F B n K ( z, ξ ) = q nn +1 F B n B ( z, ξ ) , ∀ z ∈ B n . An invariant function and some boundary asymptotics
Theorem 1.
Let (3.1) SK Ω ( z, w ) = S Ω ( z, w ) n +1 K Ω ( z, w ) n , where S Ω and K Ω are the Szegő and Bergman kernels on Ω . Then SK Ω ( z, w ) isinvariant under biholomorphic mappings satisfying the hypotheses of Proposition 2,i.e., if Φ : Ω −→ Ω is such a mapping then we have SK Ω ( z, w ) = SK Ω (Φ( z ) , Φ( w )) . Proof.
It is a well-known fact (see for example section 6.1 in [7]) that(3.2) K Ω ( z, w ) = (det J C Φ( z )) K Ω (Φ( z ) , Φ( w )) (cid:16) det J C Φ( w ) (cid:17) . Hence from (2.2) and (3.2), we get S Ω ( z, w ) n +1 K Ω ( z, w ) n = S Ω (Φ( z ) , Φ( w )) n +1 (det J C Φ( z )) n (cid:16) det J C Φ( w ) (cid:17) n K Ω (Φ( z ) , Φ( w )) n (det J C Φ( z )) n (cid:16) det J C Φ( w ) (cid:17) n = S Ω (Φ( z ) , Φ( w )) n +1 K Ω (Φ( z ) , Φ( w )) n . N THE SZEGŐ METRIC 6 (cid:3)
Remark . One can easily calculate SK B n ( z, z ) , where B n is the unit ball in C n , using(2.4) and the well known formula K B n ( z, w ) = n ! π n − z · w ) n +1 for the Bergman kernel on the unit ball to obtain SK B n ( z, z ) = 1 c n +1 n ( n − nπ ) n , ∀ z ∈ B n . For the remainder of this section we assume that the defining function ρ for Ω hasbeen chosen to satisfy Fefferman’s approximate Monge-Ampère equation − det (cid:18) ρ k ρ j ρ jk (cid:19) ≤ j,k ≤ n = 1 + O (cid:0) | ρ | n +1 (cid:1) (see [10] – we could also use the not-completely-smooth exact solution to this equation[6, 17]).We set r = − ρ ; thus r > in Ω .We have the following asymptotic expansions of the Bergman and Szegő kernels(see [9, 12, 14] and additional references cited in these papers, but the material weare quoting is set forth especially clearly in section 1.1 and Lemma 1.2 from [15]): K Ω ( z, z ) = ( n ! π n r n +1 + ( n − · q Ω r n − + O (cid:0) r n − (cid:1) , n ≥ π r + 3˜ q Ω · log r + O (1) , n = 2 S Ω ( z, z ) = ( n − c n π n r n + ( n − · q Ω c n r n − + O (cid:0) r n − (cid:1) , n ≥ c π r + q Ω c r + O ( | log r | ) , n = 3 c π r + µ + ˜ q Ω c · r log r + O ( r ) , n = 2 , where µ ∈ C ∞ (Ω) and q Ω and ˜ q Ω are certain local geometric boundary invariants – interms of Moser’s normal form [8] we have q Ω = π n (cid:13)(cid:13) A (cid:13)(cid:13) for n ≥ and ˜ q Ω = − π A for n = 2 . Moreover, r n +1 K Ω ( z, z ) ∈ C n +1 − ǫ (cid:0) Ω (cid:1) and r n S Ω ( z, z ) ∈ C max { n, }− ǫ (cid:0) Ω (cid:1) for each ǫ > . (The remainder terms are equal to a power of r times a first-degreepolynomial in log r with coefficients in C ∞ (Ω) ; later in this section the remainderterms have a similar structure but with higher degree in log r .)Combining these results we obtain the following. Theorem 2.
The function SK Ω ( z, z ) satisfies SK Ω ( z, z ) ∈ ( C n − ǫ (cid:0) Ω (cid:1) , n ≥ C − ǫ (cid:0) Ω (cid:1) , n = 2 N THE SZEGŐ METRIC 7 with asymptotics SK Ω ( z, z ) = ( n − c n +1 n ( nπ ) n + ( n − · q Ω c n +1 n n n r + O ( r ) , n ≥ c (3 π ) + q Ω c · r + O ( r | log r | ) , n = 3 c π + µ r + µ r log r + π ˜ q c r log r + O ( r log r ) , n = 2 for z close to the boundary, where µ , µ ∈ C ∞ (Ω) . We will use this result to examine the relation between the Bergman and Szegőmetrics. It will be helpful to introduce the quantity E ( z, ξ ) = ( n + 1) (cid:0) F Ω S ( z, ξ ) (cid:1) − n (cid:0) F Ω B ( z, ξ ) (cid:1) . Theorem 3.
For n ≥ the following hold. (a) E ∈ C n − − ǫ (cid:0) T Ω (cid:1) . (b) There are constants < m Ω < M Ω < ∞ so that m Ω F Ω S ( z, ξ ) ≤ F Ω B ( z, ξ ) ≤ M Ω F Ω S ( z, ξ ) on T Ω . (c) E ( z, ξ ) = 0 when z ∈ ∂ Ω and ξ lies in the maximal complex subspace of T z ∂ Ω . (d) E ( z, ξ ) ≡ on all of T ( ∂ Ω) if and only if the boundary is locally spherical. (e) If Ω is simply connected then E ( z, ξ ) ≡ on T Ω if and only if Ω is biholo-morphic to the ball.Proof. We start by noting that(3.3) E ( z, ξ ) = n X j,k =1 ∂ (log SK Ω ( z, z )) ∂z j ∂z k ξ j ξ k . Then (a) follows from the smoothness result in Theorem 2. Statement (b) thenfollows from (a) and (2.3) along with the Bergman version of (2.3).For z ∈ ∂ Ω we use (3.3) and Theorem 2 to conclude that E ( z, ξ ) = 6 π n q Ω ( n − n − n X j,k =1 r j r k ξ j ξ k and thus E ( z, ξ ) = 0 when P nj =1 r j ξ j = 0 , verifying (c). From the same computationwe see that E will vanish on all of T ( ∂ Ω) if and only if the invariant q Ω vanishesidentically, so from Corollary 2.5 in [5] it follows that (d) holds.The “if” half of (e) follows from (2.8) and the invariance properties. The “onlyif” half follows from (d) along with Theorem C in [7] (see also [18] and section 8 of[4]). (cid:3) For n = 2 we have instead the following result. Theorem 4.
For n = 2 the following hold. (a) E ∈ C − ǫ (cid:0) T Ω (cid:1) . N THE SZEGŐ METRIC 8 (b)
There are constants < m Ω < M Ω < ∞ so that m Ω F Ω S ( z, ξ ) ≤ F Ω B ( z, ξ ) ≤ M Ω F Ω S ( z, ξ ) on T Ω . (c) If E ∈ C (cid:0) T Ω (cid:1) then the boundary is locally spherical. (d) If Ω is simply connected then E ∈ C (cid:0) T Ω (cid:1) if and only if Ω is biholomorphicto the ball, in which case we in fact have E ( z, ξ ) ≡ on T Ω .Proof. We need to explain part (c), everything else falling into place as before.If E ∈ C (cid:0) T Ω (cid:1) then the r log r term from the expansion in Theorem 2 mustdisappear, forcing ˜ q Ω ≡ . Using an argument of Burns appearing as Theorem 3.2in Graham’s paper [12] along with the previously cited material from [15] we obtainrevised expansions K Ω ( z, z ) = 2 π r + µ + 9 q ∗ Ω r log r + O ( r ) S Ω ( z, z ) = 1 c π r + µ + 1 c q ∗ Ω r log r + O ( r ) SK Ω ( z, z ) = 1 c π + µ r − q ∗ Ω c r log r + O ( r ) E ( z, ξ ) = µ − q ∗ Ω r log r n X j,k =1 r j r k ξ j ξ k + O ( r ) , where q ∗ Ω = π | A | and µ , µ , µ ∈ C ∞ (Ω) , µ ∈ C ∞ ( T Ω) . Our smoothnessassumption on E now forces q ∗ Ω ≡ and this in turn implies that the boundary isspherical. (We note that by Proposition 1.9 in [12], the condition ˜ q Ω ≡ alone doesnot guarantee that the boundary is spherical.) (cid:3) For the sake of completeness we also record the corresponding results in one di-mension.
Theorem 5.
For n = 1 the following hold. (a) E ∈ C ∞ (cid:0) T Ω (cid:1) . (b) There are constants < m Ω < M Ω < ∞ so that m Ω F Ω S ( z, ξ ) ≤ F Ω B ( z, ξ ) ≤ M Ω F Ω S ( z, ξ ) on T Ω . (c) If Ω is simply connected then E ( z, ξ ) ≡ .Proof. (a) follows from Theorem 23.2 in [2] and the well-known fact that rS Ω ( z, z ) and r K Ω ( z, z ) are in C ∞ (cid:0) Ω (cid:1) and are nowhere vanishing on Ω . Statement (b) followsfrom (a) as in the proof of Theorem 3 above.(c) follows from (2.8), invariance properties and the Riemann mapping theorem. (cid:3) N THE SZEGŐ METRIC 9 Comparison with the Carathéodory metric
In this section we discuss the comparison between the Carathéodory and Szegőmetrics and show that the Szegő metric is always greater than or equal to theCarathéodory metric. The proof follows the same method that was used to showthat the Bergman metric is greater than or equal to the Carathéodory metric in [13].We define the Carathéodory metric on a domain Ω ⊂ C n at p ∈ Ω in the direction ξ ∈ C n , F Ω C ( p, ξ ) , as F Ω C ( p, ξ ) = sup n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂φ ( p ) ∂z j ξ j (cid:12)(cid:12)(cid:12)(cid:12) ! / : φ ∈ O (Ω , ∆) , φ ( p ) = 0 , where O (Ω , ∆) denotes the set of holomorphic mappings from Ω to ∆ , the unit discin C . Theorem 6.
The Szegő metric is greater than or equal to the Carathéodory metric.Proof.
One can show that(4.1) (cid:0) F Ω S ( p, ξ ) (cid:1) = sup n | ξg ( p ) | : g ∈ H ( ∂ Ω) , g ( p ) = 0 , k g k L ( ∂ Ω) = 1 o S ( p, p ) using the Hilbert space method. Refer to Theorem 6.2.5 in [16] for further details.Let p ∈ Ω . We have k S ( · , p ) k L ( ∂ Ω) = k S ( p, · ) k L ( ∂ Ω) = Z ∂ Ω S ( p, ζ ) S ( p, ζ ) dσ F ( ζ ) = S ( p, p ) = S ( p, p ) . Let φ : Ω −→ ∆ be a holomorphic function with φ ( p ) = 0 . Define a holomorphicfunction g : Ω −→ ∆ as follows: g ( z ) = S ( z, p ) p S ( p, p ) φ ( z ) . Then k g k L ( ∂ Ω) ≤ and g ( p ) = 0 . Hence from (4.1) we get(4.2) (cid:0) F Ω S ( p, ξ ) (cid:1) ≥ | ξg ( p ) | S ( p, p ) = S ( p, p ) | ξφ ( p ) | S ( p, p ) = | ξφ ( p ) | . Therefore we get F Ω S ( p, ξ ) ≥ F Ω C ( p, ξ ) . (cid:3) Remark . This argument works on any smoothly bounded pseudoconvex domainwhere the Szegő metric is defined.
Remark . The equation (2.8) shows that the inequality F Ω S ( p, ξ ) ≥ F Ω C ( p, ξ ) is sharpeven in some cases where F Ω C ( p, ξ ) > , whereas we have F Ω B ( p, ξ ) (cid:13) F Ω C ( p, ξ ) if F Ω C ( p, ξ ) > [16]. N THE SZEGŐ METRIC 10 Comparison with the Bergman metric
In this section we carry out some computations on annuli to show that the con-stants m Ω and M Ω in Theorem 5 must depend on Ω . Theorem 7.
There are no constants < m < M < ∞ independent of Ω with theproperty that m F Ω S ( z, ξ ) ≤ F Ω B ( z, ξ ) ≤ M F Ω S ( z, ξ ) on T Ω .Proof. The results of Proposition 4 below show that F Ω r S ( √ r, /F Ω r B ( √ r, → ∞ and F Ω r S ( √ r, /F Ω r B ( √ r, → as r → , where Ω r = { z ∈ C : r < | z | < } . (cid:3) Proposition 4.
Let Ω r = { r < | z | < } ⊂ C and r ∈ (0 , . We have lim r → F Ω r B ( √ r, p log(1 /r ) = 2 and lim r → √ r · F Ω r S ( √ r,
1) = 12 . Also, lim r → F Ω r B ( √ r, p log(1 /r ) = √ and lim r → F Ω r S ( √ r,
1) = 1 . Proof.
On the boundary of a planar domain, the Fefferman measure is c ds , where ds denotes the element of arclength. In view of Remark 3, we may set c = 2 so that dσ F = ds .The Szegő and Bergman spaces of Ω r admit orthonormal bases { a n ( r ) z n } n ∈ Z and { b n ( r ) z n } n ∈ Z with a n ( r ) and b n ( r ) ≥ ; thus B r ( z, ζ ) = P n ∈ Z ( b n ( r )) z n ζ n and S r ( z, ζ ) = P n ∈ Z ( a n ( r )) z n ζ n . One can calculate a n ( r ) and b n ( r ) as follows: Z ∂ Ω r | a n ( r ) z n | ds = Z | z | = r r n | a n ( r ) | ds + Z | z | =1 | a n ( r ) | ds = | a n ( r ) | π ( r n +1 + 1) = 1 , hence | a n ( r ) | = 12 π (1 + r n +1 ) , n ∈ Z . N THE SZEGŐ METRIC 11
Also we have Z Ω r | b n ( r ) z n | dA = Z π Z r | b n ( r ) | t n t dt dθ = | b n ( r ) | π n + 2 (1 − r n +2 ) = 1 , n = − , Z Ω r | b − ( r ) z − | dA = Z π Z r | b − ( r ) | t dt dθ = | b − ( r ) | π ln(1 /r ) = 1 , and so | b n ( r ) | = ( n +1 π · − r n +2 , n = − , π ln(1 /r ) , n = − . Let B r ( z, ζ ) and S r ( z, ζ ) be the Bergman and Szegő kernel on Ω r respectively and z ∈ Ω r . We have (cid:0) F Ω r B ( z, (cid:1) = ∂∂ log B r ( z, z ) ∂z ∂z = B r ( z, z ) · ( B r ( z, z )) zz − | ( B r ( z, z )) z | ( B r ( z, z )) = β ( z, r ) · β ( z, r ) − | β ( z, r ) | ( β ( z, r )) , where β ( z, r ) = B r ( z, z ) , β ( z, r ) = ( B r ( z, z )) z , and β ( z, r ) = ( B r ( z, z )) zz . We also get (cid:0) F Ω r S ( z, (cid:1) = α ( z, r ) · α ( z, r ) − | α ( z, r ) | ( α ( z, r )) , where α ( z, r ) = S r ( z, z ) , α ( z, r ) = ( S r ( z, z )) z , and α ( z, r ) = ( S r ( z, z )) zz . Let us calculate α j ( r q , r ) for j = 0 , , , q > and estimate F Ω r S ( r q , : π α ( r q , r ) = X n ∈ Z r n +1 ) r nq , π α ( r q , r ) = X n ∈ Z r n +1 ) · n · r (2 n − q , π α ( r q , r ) = X n ∈ Z r n +1 ) · n · r n − q . N THE SZEGŐ METRIC 12
Note that π α ( √ r, r ) = 21 + √ r + 2 r r + O (cid:0) r (cid:1) , π α ( √ r, r ) = − √ r (1 + r ) − √ r r + O (cid:0) r / (cid:1) , πα ( √ r, r ) = 1 r (1 + r ) + 51 + r + O ( r ) , and that π α ( √ r, r ) = 11 + r + r / r + O (cid:0) r / (cid:1) , π α ( √ r, r ) = r / r − r / r + O (cid:0) r / (cid:1) , π α ( √ r, r ) = 11 + r + r / r + O (cid:0) r / (cid:1) , which one can verify easily using the comparison test with the geometric series.Therefore we get lim r → r · (cid:0) F Ω r S ( √ r, (cid:1) = 14 , and lim r → (cid:0) F Ω r S ( √ r, (cid:1) = 1 . One can calculate β j ( r q , r ) ’s for j = 1 , , and estimate F Ω r B ( r q , in a similarway: π β ( r q , r ) = X n ∈ Z \{− } ( n + 1)1 − r n +2 r nq + 12 r q log (1 /r ) ,π β ( r q , r ) = X n ∈ Z \{− } ( n + 1)1 − r n +2 · n · r (2 n − q − r q log (1 /r ) ,π β ( r q , r ) = X n ∈ Z \{− } ( n + 1)1 − r n +2 n r (2 n − q + 12 r q log (1 /r ) . We have π β ( √ r, r ) = 12 r log (1 /r ) + 21 − r + O ( r ) ,π β ( √ r, r ) = − r / log (1 /r ) − √ r (1 − r ) + O (cid:0) √ r (cid:1) ,π β ( √ r, r ) = 12 r log (1 /r ) + 4 r (1 − r ) + O (1) , N THE SZEGŐ METRIC 13 and π β ( √ r, r ) = 12 r / log (1 /r ) + 11 − r + O (cid:0) r / (cid:1) ,π β ( √ r, r ) = − r / log (1 /r ) + 2 r / − r + O (cid:0) r / (cid:1) ,π β ( √ r, r ) = 12 r / log (1 /r ) + 21 − r + O (cid:0) r / (cid:1) . Therefore we get lim r → (cid:0) F Ω r B ( √ r, (cid:1) log (1 /r ) = 4 and lim r → (cid:0) F Ω r B ( √ r, (cid:1) ) log (1 /r ) = 2 . (cid:3) Remark . We note that the Szegő and Bergman kernels of Ω r can be written inclosed form in terms of elliptic functions (see for example [3]) though that is notparticularly helpful for the computations above. References [1] Barrett, D. A floating body approach to Fefferman’s hypersurface measure.
Math. Scand. , 98,69–80, 2006.[2] Bell, S. The Cauchy transform, potential theory, and conformal mapping. CRC Press, BocaRaton, 1992.[3] Burbea, J. Effective methods of determining the modulus of doubly connected domains.
J. Math.Anal. Appl. , 62, 236–242, 1978.[4] Burns, D. and Shnider, S. Spherical hypersurfaces in complex manifolds.
Invent. Math.
33 223-246, 1976.[5] Burns, D. and Shnider, S. Real hypersurfaces in complex manifolds.
Proc. Sympos. Pure Math
Vol. XXX, Part 2, (Amer. Math. Soc.) 141–168, 1977.[6] Cheng, S. Y., Yau, S. T. On the existence of a complex Kähler metric on non-compact complexmanifolds and the regularity of Fefferman’s equation.
Comm. Pure Appl. Math.
33, 507Ð544,1980.[7] Chern, S., Ji, S. On the Riemann mapping theorem.
Ann. of Math. (2), 144, 421–439, 1996.[8] Chern, S.S., Moser, J.K. Real hypersurfaces in complex manifolds.
Acta Math.
Invent. math. , 26, 1–65, 1974.[10] Fefferman, C. Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvexdomains.
Ann. of Math. (2), 103, 395-416, 1976.[11] Fefferman, C. Parabolic invariant theory in complex analysis.
Adv. in Math. , 31, 131–262, 1979.[12] Graham, C. R. Scalar boundary invariants and the Bergman kernel.
Lecture Notes in Mathe-matics (Springer) 1276, 108–135, 1987.[13] Hahn, K. T. Inequality between the Bergman metric and Carathéodory differential metric.
Proc. Amer. Math. Soc. , 68, 193–194, 1978.[14] Hirachi, K. A link between the asymptotic expansions of the Bergman kernel and the Szegőkernel.
Adv. Stud. Pure Math. , 42, 115–121, 2004.[15] Hirachi, K., Komatsu, G., Nakazawa, N. Two methods of determining local invariants in theSzegő kernel,
Lecture Notes in Pure and Appl. Math. (Dekker) 143, 77-96, 1993.
N THE SZEGŐ METRIC 14 [16] Pflug, P., Jarnicki, M. Invariant distances and metrics in complex analysis. Walter de Gruyter& Co., Berlin, 1993.[17] Lee, J., Melrose, R. Boundary behavior of the complex Monge-Ampère equation.
Acta Math. ,148, 159-192, 1982.[18] Pinčuk, S. I. The analytic continuation of holomorphic mappings. (Russian)
Mat. Sb. (N.S.)98, 416-435, 1975; English trans.
Math USSR Sb.
27, 375-392, 1975.27, 375-392, 1975.