An Estimate for the Squeezing function and estimates of invariant metrics
aa r X i v : . [ m a t h . C V ] N ov AN ESTIMATE FOR THE SQUEEZING FUNCTION ANDESTIMATES OF INVARIANT METRICS
J. E. FORNÆSS AND ERLEND F. WOLD
Abstract.
We give estimates for the squeezing function on strictlypseudoconvex domains, and derive some sharp estimates for the Carath´eodory,Sibony and Azukawa metrics near their boundaries. Introduction
Let Ω be a bounded domain in C n . The squeezing function [1] measureshow much a domain looks like the unit ball observed from a given point z .More precisely it is defined as follows: For a given injective holomorphicmap f : Ω → B n satisfying f ( z ) = 0 we set S Ω ,f ( z ) := sup { r > r B n ⊂ f (Ω) } , and then we set S Ω ( z ) := sup f { S Ω ,f ( z ) } , where f ranges over all injective holomorphic maps f : Ω → B n with f ( z ) =0. It was proved in [1] that lim z → b Ω S Ω ( z ) = 1if Ω is a C -smooth strictly pseudoconvex domain, and it was proved in [3]that the squeezing function is bounded on any bounded convex domain. Ourgoal is to improve this estimate when the boundary has higher regularity,and to give an application to invariant metrics. Theorem 1.1.
Let
Ω = { δ < } ⊂ C n be a strictly pseudoconvex domainwith a defining function δ of class C k for k ≥ . The squeezing function S Ω ( z ) for Ω satisfies the estimate S Ω ( z ) ≥ − C · p | δ ( z ) | for a fixed constant C . If we even have k ≥ , then there exists a constant C > such that the squeezing function S Ω ( z ) for Ω satisfies S Ω ( z ) ≥ − C · | δ ( z ) | for all z Date : July 16, 2018.
Combining with a theorem due to D. Ma [4] and a result of Deng, Guanand Zhang [1], an immediate consequence is a sharp estimate for invariantmetrics near the boundary of a strictly pseudoconvex domain. Before westate the result, we briefly recall the definitions of some invariant metrics.Let ∆ denote the unit disc, and let O ( M, N ) denote the holomorphic mapsfrom M to N. • Kobayashi metric K Ω ( p, ξ ) . We define K Ω ( p, ξ ) = inf {| α | ; ∃ f ∈ O (∆ , Ω) f (0) = p, αf ′ (0) = ξ } . • Carath´eodory metric C Ω ( p, ξ ) . We define C Ω ( p, ξ ) = sup {| f ′ ( p )( ξ ) | ; ∃ f ∈ O (Ω , ∆) f ( p ) = 0 } . • Sibony metric S Ω ( p, ξ ) . We define S Ω ( p, ξ ) = sup { ( P i,j ∂ u ( p ) ∂z i ∂z j ξ i ξ j ) / , u ( p ) = 0 , ≤ u < , u is C near p and ln u is plurisubharmonic in Ω } . • Azukawa metric A U ( p, ξ ) . We define A Ω ( p, ξ ) = sup u ∈ P Ω ( p ) { lim sup λ ց | λ | u ( p + λξ ) } where P Ω ( p ) = { u : Ω → [0 , , ln u is plurisubharmonic and ∃ M u > , r u > B n ( p, r ) ⊂ Ω , u ( z ) ≤ M k z − p k , z ∈ B n ( p, r ) } Theorem 1.2.
Let Ω ⊂ C n be a strictly pseudoconvex domain of class C , let p ∈ b Ω , and let δ be a defining function for Ω near p , such that k∇ δ ( z ) k = 1 for all z ∈ b Ω . Then if F Ω ( z, ζ ) is either the Carath´eodory,Sibony or Azukawa metric, there exists a constant C > such that (1 − C p | δ ( z ) | ) (cid:20) L π ( z ) ( ξ T ) | δ ( z ) | + k ξ N k δ ( z ) (cid:21) / ≤ F Ω ( z, ξ ) ≤ (1 + C p | δ ( z ) | ) (cid:20) L π ( z ) ( ξ T ) | δ ( z ) | + k ξ N k δ ( z ) (cid:21) / for all z near p , and all ξ = ξ N + ξ T , where π is the orthogonal projection to b Ω , ξ N is the complex normal component of ξ at π ( z ) and ξ T is the complextangential component, and L is the Levi form of δ . Ma’s result is the corresponding statement for the Kobayashi metric, andthe result is sharp in the sense that one cannot in general do better thanthe square root of the boundary distance. Proof of Theorem 1.2
The following was proved in [1], and we include the proof for the benefitof the reader.
Lemma 2.1.
Let Ω be any bounded domain in C n , and let F Ω ( z, ξ ) be eitherthe Carath´eodory, Sibony or Azukawa metric. Then S Ω ( z ) K Ω ( z, ξ ) ≤ F Ω ( z, ξ ) ≤ K Ω ( z, ξ ) for all z ∈ Ω and all ξ ∈ C n , where K denotes the Kobayashi metric.Proof. It is well known that K dominates F so we need to show the lowerestimate. Let f : Ω → B n be injective holomorphic with f ( z ) = 0, such that B r ⊂ f (Ω) where r = S Ω ( z ). For the existence of f see [1] (alternativelyone can use a limiting argument). We get that F Ω ( z, ξ ) = F f (Ω) (0 , f ∗ ξ ) ≥ F B n (0 , f ∗ ξ ) = K B n (0 , f ∗ ξ )= S Ω ( z ) K B r (0 , f ∗ ξ ) ≥ S Ω ( z ) K f (Ω) (0 , f ∗ ξ ) = S Ω ( z ) K Ω ( z, ξ ) . (cid:3) Proof of Theorem 1.2:
By Lemma 2.1 we have that S Ω ( z ) K Ω ( z, ξ ) ≤ F Ω ( z, ξ ) ≤ K Ω ( z, ξ )Then combining Theorem 1.1 with the fact that Theorem 1.2 holds with F Ω ( z ) replaced by K Ω ( z ) (see [4]) completes the proof. (cid:3) Proof of Theorem 1.1
The following provides the key geometric setup for the proof. Let k = 3or 4, and let Ω be a bounded strongly pseudoconvex domain of class C k . Lemma 3.1.
Let p ∈ b Ω . There exists an injective holomorphic map φ :Ω → C n such that ˜Ω = φ (Ω) satisfies the following: (i) ˜Ω ⊂ B n , (ii) φ ( p ) = (1 , , · · · ,
0) =: a and φ − ( b B n ) = { p } , (iii) near a we have that, ˜Ω = { ρ < µ } , < µ < where ρ ( z ) = | z − (1 − µ ) | + k z ′ k + O ( | z − | ) + O ( k z − a k k ) . Proof.
By the main theorem in [2] there exists a map φ such that (i) and (ii)are satisfied. That we can achieve (iii) follows from the proof which consistsof three steps. We first apply an automorphism of C n to ensure that, locallynear p = 0, our domain has a defining function ρ ( z ) = 2 Re ( z ) + k z k + O ( k z k k ) . (3.1)To achieve this one approximates a local map with jet interpolation usingthe Anders´en-Lempert theory. We next apply another automorphism of C n which can be chosen to match the identity at the origin to any given order,so we still have a defining function of the form (3.1). The final exposing map J. E. FORNÆSS AND ERLEND F. WOLD is of the form ϕ = φ ◦ α , where φ ( z ) = ( f ( z ) , z , ..., z n ) where f is injectiveholomorphic with f ′ (0) >
0, and α ( z ) can be chosen to match the identityto any given order at the origin. By a translation we assume that ϕ (0) = 0.We then have a defining function for ϕ (Ω) of the form ρ ( z ) = 2 Re ( c z + c z + c z ) + | c | | z | + k z ′ k + O ( | z | ) + O ( k z k k )= 2 c Re ( z ) + | c | | z | + k z ′ k + O ( | z | ) + O ( k z k k ) . Applying the linear change of coordinates ( z , z ′ ) ( z /c , z ′ ), we get adefining function ρ ( z ) = 2 Re ( z ) + | z | + k z ′ k + O ( | z | ) + O ( k z k k ) . By chosing a small 0 < µ < µϕ (Ω) is contained in thetranslated unit ball { Re ( z ) + k z k < } , with defining function ρ ( z ) = 2 µRe ( z ) + | z | + k z ′ k + O ( | z | ) + O ( k z k k ) , which is the same as (iii) when translated ( z , z ′ ) ( z + 1 , z ′ ). (cid:3) Remark 3.2. On b ˜Ω the remainder term in (iii) is actually O ( | z − | k/ ).To see this we first translate ˜Ω to the origin, set ˜ z = z − , ˜ z = (˜ z , z ′ ) sothat it is defined by˜ ρ (˜ z ) = 2 Re (˜ z ) + | ˜ z | + 1 µ k z ′ k + O ( | ˜ z | ) + O ( k ˜ z k k ) < . We estimate k z ′ k on ˜ ρ = 0. If k z ′ k ≤ | ˜ z | the remainder term is less than C | ˜ z | k = O ( | z | k/ ). If | ˜ z | ≤ k z ′ k then the remainder term is O ( k z ′ k k ) andwe get k z ′ k + O ( k z ′ k k ) = µ ( − Re (˜ z ) − | ˜ z | + O ( | ˜ z | ))= µ | ˜ z | ( − Re (˜ z ) | ˜ z | − | ˜ z | + O ( | ˜ z | ) | ˜ z | ) . This implies that the remainder term is O ( k z ′ k k ) = O ( | z − | k/ ) . From now on we assume that Ω = ˜Ω and satisfies (i)-(iii) above. Then Ωis ”almost” contained in the ball B µ ⊂ B n defined by | z | + 1 µ k z ′ k < . We will use automorphisms of the ball B n of the form φ r ( z , z ′ ) = z − r − rz , √ − r − rz z ′ ! . We have that φ r leaves B µ invariant. To prove the theorem, we will estimatetwo things:(a) How much φ r (Ω) sticks out of B µ and(b) the size of the largest ball in B µ contained in φ r (Ω) . Estimate (a).Lemma 3.3.
There exists a constant
C > such that for w ∈ bφ r (Ω) wehave that | w | + µ k w ′ k ≤ C (1 − r ) k − .Proof. We would like to express the maximum of the function k φ r ( z ) k interms of (1 − r ) on b Ω, i.e. , we look at k φ r ( z ) k = | z − r | + µ (1 − r ) k z ′ k | − rz | = | z − r | | − rz | + 1 µ (1 − r ) | z ′ | | − rz | for z ∈ b Ω. Fix any η > . We show first that if z ∈ B n with | z − | > η, then we have a uniform estimate k φ r ( z ) k ≤ C (1 − r ) . In this case we have that the denominator of the second term staysbounded independent of r , while | z ′ | ≤
1, hence the term goes to zero like(1 − r ). For the other term we write | z − r | | − rz | = 1 + (1 − r )( | z | − | − rz | ≤ C (1 − r ) . Next we look at | z − | ≤ η . If η is chosen small enough, the localdescription (iii) is valid. Hence if | z − | < η and if z ∈ b Ω we have that k z ′ k = −| z − (1 − µ ) | + O ( | z − a | k/ ) + µ = −| z − | − µRe ( z −
1) + O ( | z − a | k/ ) , which gives that1 µ k z ′ k = − µ | z − | − Re ( z −
1) + O ( | z − a | k/ ) ≤ −| z − | − Re ( z −
1) + O ( | z − a | k/ )= 1 − | z | + O ( | z − a | k/ ) . J. E. FORNÆSS AND ERLEND F. WOLD
Hence | z − r | + µ (1 − r ) k z ′ k | − rz | = | z − r | + (1 − r )(1 − | z | ) | − rz | + (1 − r ) O ( | z − | k/ ) | − rz | = 1 + (1 − r ) O ( | z − | k/ ) | − rz | ≤ C (1 − r ) | − rz | k/ | − rz | ≤ C − r | − rz | − ( k/ ≤ C − r (1 − r ) − ( k/ ≤ C (1 − r ) k − . (cid:3) Estimate (b).
We define B µη, ˜ η = {| z − (1 − η ) | + ˜ ηµ | z ′ | < η } with constants 0 < η ≤ ˜ η < η. Lemma 3.4.
We set ˜ η = ( η, k = 4 η − Cη , k = 3 (i) If k = 4 then B µη, ˜ η ⊂ Ω for all η small enough(ii) If k = 3 , and the constant C > is fixed large enough, then B µη, ˜ η ⊂ Ω for all η small enough.Proof. For η small enough, the ellipsoid B µη, ˜ η is contained in the region wherethe local defining function ρ is defined. Since ρ is plurisubharmonic it sufficesto show that ρ ≤ bB µη, ˜ η . We translate coordinates, by setting ˜ z = z − z = (˜ z , z ′ ) . We want to show that { ηRe (˜ z ) + | ˜ z | + ˜ ηµ | z ′ | = 0 } is contained in the set { µRe (˜ z ) + | ˜ z | + k z ′ k + O ( | ˜ z | ) + O ( k ˜ z k k ) ≤ } . Write ˜ z = x + iy . On the boundary of the ellipsoid we have that2 ηx + x + y + ˜ ηµ k z ′ k = 0 ⇔ µ ˜ η y + k z ′ k = − µ ˜ η (2 ηx + x ) and consequently we get on the boundary of the ellipsoid that k ˜ z k = x + y + k z ′ k ≤ x + µ ˜ η y + k z ′ k = x − µ ˜ η (2 ηx + x )= − x ( − x + µ ˜ η (2 η + x ))It follows that k ˜ z k ≤ C | x | , and so k ˜ z k k ≤ C | x | k/ (3.2)Consider again the boundary of the ellipsoid; we have x + y + 2 ηx + ˜ ηµ k z ′ k = 0Hence k z ′ k = − µ ˜ η ( x + y + 2 ηx )Therefore2 µx + | ˜ z | + k z ′ k + O ( | ˜ z | ) + O ( k ˜ z k k ) ≤ µx + | ˜ z | − µ ˜ η ( | ˜ z | + 2 ηx ) + C | x | k/ + C | ˜ z | . using (3.2). It suffices therefore to show that the right side is ≤ . Thismeans: 2 µx (1 − η ˜ η ) + | ˜ z | (1 − µ ˜ η ) + C | x | k/ + C | ˜ z | ≤ . (3.3)Observe that ≤ η ˜ η ≤
1, so 1 − η ˜ η ≥ . Morever x ≤ ≤ . It suffices therefore that | ˜ z | (1 − µ ˜ η ) + C | ˜ z | k/ ≤ , (3.4)where we merged the constants C and C . When k = 4, this holds as soonas η is small enough. When k = 3, this holds when C | ˜ z | / ≤ µ ˜ η | ˜ z | / | ˜ z | / (1 − ˜ ηµ )or C ≤ | ˜ z | / ˜ η ( µ − ˜ η )This holds when | ˜ z | ≥ ˜ Cη for large enough ˜ C. To complete the proofwe need to consider the case when k = 3 and | ˜ z | ≤ ˜ Cη , and we go backto consider the full expression (3.3). Since the sum | ˜ z | (1 − µ ˜ η ) + C | ˜ z | isnegative when η is small, it is enough to determine when2 µx (1 − η ˜ η ) + C | x | / ≤ . J. E. FORNÆSS AND ERLEND F. WOLD or equivalently when2 µx (1 − η ˜ η ) ≤ C x | x | / ⇔ µ (1 − η ˜ η ) ≥ C | x | / . By our assumption we now have that C | x | / ≤ C ( ˜ Cη ) / = C η , andso we need that 2 µ (1 − η ˜ η ) ≥ C η. Hence the choice ˜ η = η − C µ η works. (cid:3) Now let ψ ( z , z ′ ) = ( z , √ µ z ′ ). Then ψ ( B µη, ˜ η ) is the ellipsoid B η, ˜ η = {| z − (1 − η ) | + ˜ η k z ′ k < η } , Lemma 3.5.
Let < η, r < and ˜ η > . If z ∈ bB η, ˜ η , then k φ r ( z , z ′ ) k = 1 + (1 − r ) | z − | | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | + (1 − r )2(1 − η ˜ η )( Re ( z ) − | − rz | Proof. k φ r ( z , z ′ ) k = | z − r | + (1 − r ) | z ′ | | − rz | = | z − r | + (1 − r )(1 / ˜ η )( η − | z − (1 − η ) | ) | − rz | = | z − r | + (1 − r )(1 / ˜ η )( η − | z − | − ηRe ( z − − η ) | − rz | = | z − r | + (1 − r )(1 / ˜ η )( − ηRe ( z − − | z − | ) | − rz | = 1 + | z − r | − | − rz | − (1 − r ) η ˜ η Re ( z − | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | = 1 + | z | − rRe ( z ) + r − (1 − rRe ( z ) + r | z | ) | − rz |− (1 − r ) η ˜ η Re ( z − | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | = 1 + (1 − r )( | z | − ηη Re ( z ) + ( η ˜ η − | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | = 1 + (1 − r ) | z − | | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | + (1 − r )2(1 − η ˜ η )( Re ( z ) − | − rz | (cid:3) Lemma 3.6.
Let ψ ( z ) = ( z , √ µ z ′ ) . Suppose that < η, r < , − η < r and ˜ η > . Then ψ ( φ r ( B µη, ˜ η )) contains the ball of radius r − − r ) 1˜ η − | − η ˜ η | . Proof.
Since 1 − η < r, we have that 0 ∈ ψ ( φ r ( B µη, ˜ η )) . Hence it sufficesto show that k ψ ( φ r )( z ) k ≥ − − r ) η − | − η ˜ η | on the boundary of B µη, ˜ η . (This is nonempty if the expression on the right is nonnegative.) Since ψ ◦ φ r = φ r ◦ ψ , it suffices to show that k φ r ( z ) k ≥ − − r ) η − | − η ˜ η | on the boundary of B η, ˜ η . From the previous lemma we have that k φ r ( z ) k ≥ − r ) | z − | | − rz | − (1 − r )(1 / ˜ η ) | z − | | − rz | + (1 − r )2(1 − η ˜ η )( Re ( z ) − | − rz | ≥ − (1 − r )(1 / ˜ η ) | z − | | − rz | − (1 − r )2 | − η ˜ η || Re ( z ) − || − rz | ≥ − (2(1 − r ))(1 / ˜ η ) | rz − | | − rz | − (2(1 − r ))2 | − η ˜ η || rz − || − rz | ≥ − (1 − r ))(2 / ˜ η ) − | − rz | )(1 − η ˜ η ) | − rz |≥ − (1 − r ))(2 / ˜ η ) − | − η ˜ η | (cid:3) We prove Theorem 1.1
Proof.
We will estimate the squeezing function at points ( r,
0) when r < . That this gives the uniform constant claimed in Theorem 1.1,follows from the dependence on p as p varies over the boundary of theoriginal domain. In particular, the constants in our estimates can be chosenindependently of the point p , and the radial lines will foliate a neighborhoodof the boundary so that we get an estimate for all points near the boundary.The map ψ ◦ φ r maps ( r,
0) to the origin. We estimate the image of Ω . It follows from Lemma 3.3 that there exists a constant
C > w ∈ bφ r (Ω) we have that | w | + µ k w ′ k ≤ C (1 − r ) k − . Since the leftside is plurisubharmonic, the same estimate holds by the maximum principleon φ r (Ω) . Suppose that ( z , z ′ ) ∈ ψ ( φ r (Ω)) . Then ( z , z ′ ) = ψ ( w , w ′ ) =( w , √ µ w ′ ) for some w ∈ φ r (Ω) . Hence k z k = | w | + µ k w ′ k ≤ C (1 − r ) k − . It follows that ψ ( φ r (Ω)) is contained in the ball centered at the originof radius 1 + C (1 − r ) k − .We next estimate the radius of the largest ball contained in ψ ( φ r (Ω)) . ByLemma 3.4 we have ellipsoids B µη, ˜ η = {| z − (1 − η ) | + ˜ ηµ | z ′ | < η } containedin Ω for certain η, ˜ η : We set˜ η = ( η, k = 4 η − Cη , k = 3 . (i) If k = 4 we have that B µη, ˜ η ⊂ Ω for all η small enough, and(ii) if k = 3, and the constant C > B µη, ˜ η ⊂ Ω for all η small enough. We can then estimate instead the largest ball containedin ψ ( φ r ( B µη, ˜ η )) . We use Lemma 3.6: Suppose that 0 < η, r < , − η < r and ˜ η > . Then ψ ( φ r ( B µη, ˜ η )) contains the ball of radius r − − r ) 1˜ η − | − η ˜ η | . We deal first with the case k = 4 . Then we assume that 1 − η < r and˜ η = η. It follows that ψ ( φ r (Ω)) ⊃ ψ ( φ r (cid:16) B µη, ˜ η (cid:17) ⊃ B (0 , q − − r ) η ) . We choose a fixed η ,and let r → . We then get that for a fixed constant C ′ , ψ ( φ r (Ω)) ⊃ B (0 , − C ′ (1 − r )) . Hence we have shown that in the case k = 4 , k − = 1 , B (0 , − C ′ (1 − r )) ⊂ ψ ( φ r (Ω)) ⊂ B (0 , C (1 − r )) . Composing with the map λ ( z ) = z C (1 − r ) we obtain that λ ( ψ ( φ r ( r, B (0 , − C ′ (1 − r )1 + C (1 − r ) ) ⊂ λ ( ψ ( φ r (Ω))) ⊂ B (0 , . Hence it follows that the squeezing function at ( r,
0) is at least 1 − C ′′ (1 − r ) . Since the defining function δ ( z ) = − (1 − r ) + O (1 − r ) for z = ( r,
0) and r close to 1, we obtain Theorem 1.1 in the case when k = 4 . It remains to do the case k = 3 . It follows as above that ψ ( φ r (Ω)) is contained in the ball centered at theorigin of radius 1 + C (1 − r ) k − = 1 + C (1 − r ) .As above we suppose that 0 < η, r < , − η < r , and we have that ψ ( φ r ( B µη, ˜ η )) contains the ball of radius r − − r ) 1˜ η − | − η ˜ η | . We have that ˜ ηη = 1 − Cη , and so it follows that ψ ( φ r (Ω)) ⊃ ψ ( φ r ( B µη, ˜ η ) ⊃ B (cid:18) , r − − r ) 1˜ η − Cη (cid:19) ⊃ B (cid:18) , r − − r ) 1 η − Cη (cid:19) . In this case, we let η depend on r. Set η = √ − r . Then r = 1 − η > − η if r is close enough to 1 . We then get that ψ ( φ r (Ω)) ⊃ B (cid:18) , r − − r ) 1 η − Cη (cid:19) = B , s − − r ) 1 √ − r − C √ − r ! = B (cid:18) , q − (2 + 4 C ) √ − r (cid:19) ⊃ B (cid:0) , − (2 + 4 C ) √ − r (cid:1) Now it follows by the same scaling type argument with a map λ that weget the desired lower bound for the squeezing function in the case k = 3 . (cid:3) An example
Let Ω be the domain Ω := B n \ B n . We will show that S Ω ( z ) cannotapproach 1 faster than 1 − C dist( z, b Ω). By abuse of notation we set r =( r, , ..., , < r < a = (1 / , , ..., B n from a to r is (log( r − r ) − log(3)). Now let f : Ω → B n be an injective holomorphic map with f ( r ) = 0. Then f extends to a holomorphic map ˜ f : B n → B n , so by the decreasing propertyof the Kobayashi metric we have that the Kobayashi distance between f ( r )and f ( a ) is less that log( r − r ). It follows that S Ω ,f ( r ) ≤ r . References
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