Cheung, WS

Wong, B

Rocky Mountain Journal of Mathematics

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Title An integral inequality of an intrinsic measure on boundeddomains in CnAuthor(s) Cheung, WS; Wong, BCitation Rocky Mountain Journal of Mathematics, 1992, v. 22 n. 3, p. 825-836Issued Date 1992URL http://hdl.handle.net/10722/75376Rights Creative Commons: Attribution 3.0 Hong Kong LicenseROCKY MOUNTAINJOURNAL OF MATHEMATICSVolume 22, Number 3, Summer 1992AN INTEGRAL INEQUALITY OFAN INTRINSIC MEASURE ONBOUNDED DOMAINS IN CnWING-SUM CHEUNG AND B. WONG1. Introduction. Let D be a complete hyperbolic bounded domainin Cn in the sense of [7]. We denote by MED the differential Eisenman-Kobayashi n-measure (defined with respect to the unit ball) on D.Since we may endow D with a global coordinate system, MED cantherefore be viewed as a function. The main goal of this paper is toprove the following theorem.Theorem. If we assume there exists a neighborhood N of ∂D in Cnwhere MED satisfies the growth condition|MED (z)| ≥k(r(z))m+s,where k = positive constant, r(z) = the euclidean distance from zto ∂D, z ∈ N ∩ D, m and s positive numbers, then we can find aneighborhood U of ∂D in Cn such that for all z0 ∈ U ∪ D, wheneverthe closed disk {z0 + ρz1 : ρ ∈ C, |ρ| ≤ 1}, z1 ∈ Cn, lies in U ∩D, theinequalityln |MED (z0)| ≤nmπ∫ 2π0ln |MED (z0 + eiθz1)| dθfor |MED | holds.Typical examples satisfying conditions of our theorem include ana-lytic polyhedra, strongly pseudoconvex domains and certain domainsof holomorphy with smooth real analytic boundary [1]. For the case ofstrongly pseudoconvex domains we have the following corollary.Received by the editors on May 15, 1990.Research of second author partially supported by an NSF grant.Copyright c©1992 Rocky Mountain Mathematics Consortium825826 W. CHEUNG AND B. WONGCorollary. Let D be a strongly pseudoconvex domain in Cn withsmooth boundary. Then there exists a neighborhood U of ∂D in Cnsuch that for all z0 ∈ U ∩D, whenever the closed disk{z0 + ρz1; ρ ∈ C, |ρ| ≤ 1}, z1 ∈ Cn,lies in U ∩D, the inequalityln |MED (z0)| ≤1π∫ 2π0ln |MED (z0 + eiθz1)| dθholds.Our proof rests on the boundary assumption of the intrinsic measureand the classical Hartogs’ construction of analytic family of disks [11].It is quite clear from a minor modification of the proof that our integralinequality also holds on an analytically embedded disk (i.e., the imageof a holomorphic embedding f : B → D, here B = {z ∈ C : |z| < 1},which is homeomorphic up to the boundary ∂B with f(∂B) ⊂ D andf(0) = z0). One can see this inequality imposes a restriction on MEDwhen the analytic disk is large and sufficiently close to the boundary.This type of inequality can be generalized to other low dimensionalintrinsic measures. A sharpened result can also probably be derivedalong our line. The arrangement of our paper can be summarized asbelow.Section 2. Definition of Eisenman-Kobayashi measure. Section 3.A boundary estimate of Eisenman-Kobayashi measure on stronglypseudoconvex domain and proof of the Corollary. Section 4. Proofof our main statement.2. Definition of Eisenman-Kobayashi measure. For the basicdefinitions and a survey of this subject, one should consult [7, 8]. Sinceour Eisenman-Kobayashi n-measures are defined with respect to theball in Cn, which is somewhat different from what had been done in[7, 8], we shall include our definition here.Let N be a complex manifold of dimension n. The EisenmanKobayashi n-measure MEN is an (n, n)-form |MEN | · (dz1 ∧ dz¯1 ∧ · · · ∧INTRINSIC MEASURE ON BOUNDED DOMAINS 827dzn ∧ dz¯n) on N , such that |MEN | is defined for all x ∈ N as|MEN (x)| =inf{1R2n: ∃ f ∈ Hol (BRn , N) such that f(0) = x, det (Jf(0)) = 1},where BRn is the euclidean ball with center 0 and radius R in Cn,Hol (BRn , N) the set of all holomorphic maps from BRn to N , and Jf(0)the Jacobian matrix of f at 0.It is easy to check that for a bounded domain D in Cn, |MED (x)| = 0for all x ∈ D. |MED | is in general a semicontinuous function [12]. WhenD is a complete hyperbolic bounded domain, |MED | can be proved to becontinuous. All complete hyperbolic domains in Cn are pseudoconvex[7].3. A boundary estimate of MED on S.P.C. domains and proofof the corollary. Let D be a strongly pseudoconvex domain in Cnwith smooth boundary. The following will be proved in this section.Theorem. There exist a neighborhood U of ∂D in Cn and a positiveconstant c such that for all z ∈ U ∩D,|MED (z)| ≥c(r(z))n+1,where r(z) is the euclidean distance function from z to the boundary ofD.Note that since ∂D is compact and everywhere strongly pseudocon-vex, the above statement can be reduced to the following local problem:For all p ∈ ∂D, there exists a neighborhood V of p in Cn and apositive constant k such that for all z ∈ D ∩ V ,|MED (z)| ≥k(r(z))n+1.3.1 Localization lemma. Let D be a bounded domain in Cn, D1another domain such that D ∩D1 is nonempty.828 W. CHEUNG AND B. WONGDefinitions. (1) Let z, w belong to D. Then d(z, w) = inf {P (a, b) :∃ f ∈ Hol (Bn, D) s.t. f(a) = z, f(b) = w, where P is the Kobayashimetric on Bn}; here Bn is the unit ball in Cn.(2) Let z belong to D ∩D1, then d(z) = inf {d(z, w) : w ∈ D −D1}.Lemma A. Let us denote Dˆ = D ∩D1; then for all z ∈ Dˆ, we have|MEDˆ(z)| ≤ (coth d(z))2n · |MED (z)|,where MEDˆand MED denote the Eisenman-Kobayashi measures on Dˆand D, respectively.Proof. First of all, let us fix z ∈ Dˆ and letr = sup{r′ : ∃f ∈ Hol (Br′n , Dˆ) s.t. f(0) = z, det (Jf(0)) = 1}.Then we choose a number R which is slightly larger than r. From ourchoice of r it is obvious that there is an f ∈ Hol (BRn , D) such thatf(0) = z, det (Jf(0)) = 1 and it maps a boundary point of Brn toa point belonging to D − Dˆ. One can see that if w is such a pointbelonging to D − Dˆ, then d(z) ≤ d(z, w). From our definition of d, weobserve thatd(z) ≤ d(z, w) ≤ (1/2) ln[(1 + r/R)/(1− r/R)](distance-decreasing property under holomorphic mappings; consider fto be a holomorphic map from BRn to D [7]). Hence,1/r ≤ coth d(z) · (1/R),(1/r)2n ≤ (coth d(z))2n · (1/R)2n.This inequality is true for all the R’s satisfying the properties mentionedabove. Considering the definition of MED (z), one can now conclude ourdesired inequality|MEDˆ(z)| ≤ (coth d(z))2n · |MED (z)|.Lemma B. Suppose D is a strongly pseudoconvex bounded domainin Cn and D1 is a neighborhood of a boundary point of D. Let thisINTRINSIC MEASURE ON BOUNDED DOMAINS 829boundary point be p, Dˆ = D ∩ D1, and MEDˆ ,MED the Eisenman-Kobayashi measures of Dˆ and D, respectively. Then we havelimz→p|MED (z)||MEDˆ(z)| = 1 for all z ∈ Dˆ.Proof. We divide our proof into two steps.(1) Since the inclusion map Dˆ → D is holomorphic, by the volume-decreasing property [7] we have|MEDˆ(z)| ≥ |MED (z)| i.e.,|MEDˆ(z)||MED (z)|≥ 1.(2) From Lemma (A) we have|MEDˆ(z)| ≤ (coth d(z))2n · |MED (z)|.Now it is clear from our definitions thatd(z, w) ≥ dk(z, w) ∀z, w ∈ D,where dk is the Kobayashi metric on D [7]. However, if we setdk(z,D − Dˆ) = inf {dk(z, w) : w ∈ D − Dˆ},it is known thatlimz→p dk(z,D − Dˆ) = ∞if D is strongly pseudoconvex (this statement can easily be derived fromthe result of I. Graham [5]). Thus, d(z) will go to infinity as z ∈ Dˆapproaches p; consequently, coth d(z) will tend to 1 as z ∈ Dˆ tends top. At this point we obtain another inequality:limz→p|MEDˆ(z)||MED (z)|≤ 1.Combining these two inequalities, we thereby complete the proof of ourlocalization lemma.830 W. CHEUNG AND B. WONG3.2 Proof of our estimate. Before embarking on our proof, wefirst make two remarks here.Remark 1. An analytic ellipsoid is a strongly pseudoconvex domainA in Cn which can locally be described as: If p ∈ ∂A and W is asufficiently small open neighborhood of p in Cn, thenA ∩W = {z ∈ Cn : g(z) = −z1 − z¯1 +n∑i,j=1bijziz¯j < 0},where [bij ]ni,j=1 is a hermitian positive definite matrix.In our expression p is the origin of the coordinates {z1, z2, . . . , zn}, z1is the complex normal of ∂A at p, and {z2, . . . , zn} is the basis of themaximal complex tangent space Tp(∂A). It is known that any analyticellipsoid is biholomorphically equivalent to the unit ball in Cn, and theEisenman-Kobayashi measure on the unit ball is equal to the volumeform of the Bergman metric (with the reservation of the multiple ofa constant). The following estimate can thus be obtained from theexplicit formula of the Bergman metric on Bn (see [13], for example).There exists a sufficiently small open neighborhood W1 of ∂A in Cnand a positive constant c1 such that|MEA (z)| ≈c1(r(z))n+1∀z ∈W1 ∩A.Furthermore, by the volume-decreasing property again we have thefollowing estimate.There exists an open neighborhood W2 of p in Cn and a positiveconstant c1 such that|MEWˆ2(z)| ≥ c2(r(z))n+1∀z ∈ Wˆ2 = W2 ∩A.Remark 2. Let D be a strongly pseudoconvex boundary domain inCn with smooth boundary and p be a given boundary point of D asbefore. With a similar coordinate system {z1, z2, . . . , zn} as in Remark1, in which we can locally characterize D around p asD ∩ U1 = {z ∈ D;G(z) < 0},INTRINSIC MEASURE ON BOUNDED DOMAINS 831where U1 is a neighborhood of p in Cn, andG(z) = −z1 − z¯1 +∑aijziz¯j + 2Re{∑i≥2∂2G∂zi∂zj(p)zizj}+ O(|z|3)with respect to the above coordinate system. Since D is stronglypseudoconvex, (aij) is a hermitian positive definite matrix. We canapply our localization in Lemma B to make the assertion: For all ε > 0,there exists a neighborhood U2 of p in Cn such that∣∣∣∣∣|MEUˆ2(z)||MED (z)|− 1∣∣∣∣∣ < εfor all z ∈ Uˆ2 = U2 ∩D.Proof of our main estimate. To start, we fix a boundary point p ∈ ∂Dand choose the coordinate system {z1, . . . , zn} as in Remark 2. Nowwe construct an analytic ellipsoid As whose defining equation aroundthe point p is given bygs = −z1 − z¯1 +∑(aij − s · δij)ziz¯j ,whereδij ={1, if i = j0, if i = j,and s is a well-chosen positive constant such that As satisfies thefollowing properties:(i) there exists a neighborhood V of p in Cn such that V ∩ As ={z : gs(z) < 0};(ii) with the same V in (i), we have V ∩As ⊃ V ∩D.Graphically, our situation can be illustrated by the picture on the topof the next page.If we denote by V1 = As ∩ V (white area) and V2 = D ∩ V (shadedarea), we have by volume decreasing property|MEV2(z)| ≥ |MEV1(z)| ∀z ∈ V2.832 W. CHEUNG AND B. WONGRe(z1)VD{zi}   (i ≥ 2)AszpMoreover, if V is sufficiently small we can use Remarks 1 and 2 aboveto conclude that|MED (z)| ≥k(r(z))n+1∀z ∈ Npwhere Np = {the axis Re (z1)} ∩ V , k is a suitable constant.Finally we have to observe that all of our processes described in thissection are uniform in the following sense. We can choose a sufficientlysmall neighborhood T ⊂ ∂D of p in such a way that all the sizes ofdomains of comparison and constants can be unchanged so that all theabove arguments will remain valid for all q ∈ T . Then we further refineour V = ∪q∈T {Nq}. This completes the whole proof.Remark . More precise boundary estimates for both Eisenman-Kobayashi and Carathe´odory measures were carried out in [16] fol-lowing the original work of Graham [4,5] in the case of metrics. Thelocalization lemma in the case of the Kobayashi metric was first usedby Royden [12] and Graham [4,5]. Some other related results can befound in [13, 2, 6, 3].INTRINSIC MEASURE ON BOUNDED DOMAINS 833Proof of the Corollary. Applying our theorem, we let m = n ands = 1.4. Proof of our main statement. Take the neighborhood N as inthe assumption of our theorem. Since D is complete hyperbolic in thesense of Kobayashi, consequently ln |MED | is a continuous function on{z0 + ρz1 : ρ ∈ C, |ρ| ≤ 1} ⊂ N ∩D.We can always find a real valued function h, defined and continuouson |ρ| ≤ 1, harmonic in |ρ| < 1, and equal to (1/m) ln |MED | on |ρ| = 1,that is,h(ρ) = (1/m) ln |MED (z0 + ρz1)| ∀|ρ| = 1.Let h∗ be a harmonic conjugate of h, set g = h + ih∗. Then g iscontinuous on |ρ| ≤ 1 and holomorphic in |ρ| < 1. Next, let b beany vector in Cn with ||b|| = 1 and λ0 any real number satisfying0 < λ0 < 1. Consider the analytic diskΣλ : ρ → z0 + ρz1 + λe−g(ρ)b,in Cn where |ρ| ≤ 1 and λ fixed, 0 ≤ λ ≤ λ0.Claim. ∪0≤λ≤λ0∂Σλ ⊂⊂ D.Since for all z ∈ ∂Σλ, z is the image of some ρ with |ρ| = 1,||z − (z0 + ρz1)|| = ||λe−g(ρ)b||= λe−h(ρ) ≤ λ0e−(1/m) ln |MED (z0+ρz1)|= λ0 · |MED (z0 + ρz1)|−1/m.By our assumption, since z0 + ρz1 ∈ N ∩D, we have|MED (z0 + ρz1)| ≥k(r(z0 + ρz1))m+s.Hence,1|MED (z0 + ρz1)|1/m≤ r(z0 + ρz1) ·[(r(z0 + ρz1))sk]1/m.834 W. CHEUNG AND B. WONGMoreover, we can choose U ⊂ N to be sufficiently small so that theinequality [(r(z0 + ρz1))sk]1/m< 1also holds. Therefore, we obtain||z − (z0 + ρz1)|| < λ0 · r(z0 + ρz1).However, λ0 lies between zero and one; this yields||z − (z0 + ρz1)|| < r(z0 + ρz1),which is independent of z and λ. Hence, the inequality holds for allz ∈ ∪∂Σλ, that is, ∪∂Σλ is bounded. Furthermore, for all pointsz ∈ ∪∂Σλ and w ∈ ∂D,||z − w|| ≥ ||(z0 + ρz1)− w|| − ||z − (z0 + ρz1)||,thusr(z) ≥ r(z0 + ρz1)− ||z − (z0 + ρz1)|| > 0,hence ⋃0≤λ≤λ0∂Σλ ⊂⊂ D.This verifies our claim. Applying Kontinuita¨tssatz, we therefore obtainz0 + ρz1 + λe−g(ρ)b ∈ D ∀0 ≤ λ < 1, |ρ| ≤ 1.Hence,z0 + λe−g(0)beiθ ∈ D ∀0 ≤ λ < 1, 0 ≤ θ ≤ 2π,where b is in arbitrary direction. That is, the ball B(z0, ||λe−g(0)b||) ⊂D, for all 0 ≤ λ < 1. It implies thatB(z0, ||e−g(0)b||) ⊂ D;consequently,B(z0, e−h(0)) ⊂ D.INTRINSIC MEASURE ON BOUNDED DOMAINS 835Hence, the function f : Be−h(0)n → D (recall that b is in arbitrarydirection) such that f(z) = z0 + z is well defined and holomorphic.Note that f(0) = z0 and det (Jf(0)) = 1, hence|MED (z0)| = inf{1R2n: ∃ f ∈ Hol (BRn , D),such that f(0) = z0, det (Jf(0)) = 1}≤ 1e−2n·h(0)= e2n·h(0).Therefore,ln |MED (z0)| ≤ 2n · h(0) = 2n ·12π∫ 2π0h(eiθ) dθ=nmπ∫ 2π0ln |MED (z0 + eiθz1)| dθ,which is exactly what we want to show.REFERENCES1. Eric Bedford and J. Fornaess, A construction of peak functions on weaklypseudoconvex domains, Ann. Math. 107 (1978), 555 568.2. K. Diederich, Das Randverhalten der Bergmanschen Kernfunktion and Metrikin strong pseudokovexen Gebieton, Math. Ann. 187 (1970), 9 36.3. C. Fefferman, The Bergman kernel and biholomorphic maps of pseudoconvexdomains, Invent. Math. 26 (1974), 1 65.4. I. Graham, Boundary behavior of Carathe´odory and Kobayashi metrics onstrongly pseudoconvex domains in Cn with smooth boundary, thesis, PrincetonUniversity, Princeton, 1973.5. , Boundary behavior of Carathe´odory and Kobayashi metrics on stronglypseudoconvex domains in Cn with smooth boundary, Trans. Amer. Math. Soc. 207(1975), 219 240.6. G.M. Henkin, Analytic polyhedra are not analytically equivalent to stronglypseudoconvex domains, Soviet Math. Dokl. 14 (1973), 858 862.7. S. Kobayashi,Hyperbolic manifolds and holomorphic mappings, Marcel Dekker,New York, 1970.8. , Intrinsic distances, measures and geometric function theory, Bull.Amer. Math. Soc. 82 (1976), 357 416.9. M.H. Kwack, Generalization of the big picard theorem, Ann. Math. 90 (1969),9 22.836 W. CHEUNG AND B. WONG10. H.K. Look, Schwartz Lemma and analytic invariants, Scienta Sincica 7(1958), 453 504.11. R. Narasimhan, Several complex variables, University of Chicago Press,Illinois, 1971.12. H. Royden, Remarks on Kobayashi metric, in Several complex variables II,MD 1970, 125 137, Springer Verlag, Lecture Notes, 1971.13. E.M. Stein, Boundary behavior of holomorphic functions of several complexvariables, in Mathematical Notes, Princeton University Press, New Jersey, 1972.14. B. Wong, Biholomorphic characterization of the ball in Cn by its automor-phism group, Invent. Math. 41 (1977), 253 257.15. , On the holomorphic curvature of some intrinsic metrics, Proc. Amer.Math. Soc. 65 (1977), 57 61.16. , Boundary estimates of some intrinsic measures on strongly pseudo-convex domains, unpublished.Department of Mathematics, University of Hong Kong, Hong KongDepartment of Mathematics and Computer Science, University of Cal-ifornia, Riverside, Riverside, CA 92521

An integral inequality of an intrinsic measure on bounded domains in Cn

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An integral inequality of an intrinsic measure on bounded domains in Cn

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