Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type in

Abstract

Let be a smoothly bounded pseudoconvex domain in and assume that is a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curve , connecting points   and , and a quantity , along , which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.

1. Introduction

Let be a smoothly bounded domain in and let be a holomorphic tangent vector at a point in , and let us denote the Bergman, Caratheodory, and Kobayashi metrics at by , , and , respectively. When is a strongly pseudoconvex domain in , the optimal boundary behavior of the above metrics is well understood. For weakly pseudoconvex domains of finite type in , several authors found some results about these metrics. But in each case, the lower bounds are different from the upper bounds [1–5]. In [6], Catlin got optimal estimates in a small constant and large constant sense for pseudoconvex domains of finite type in . For pseudoconvex domains of finite type in , the first author and Herbort extended Catlin’s result to the case that the Levi-form at has corank one [7, 8] or homogeneous finite diagonal type near [9, 10].

To estimate the above invariant metrics, we need a complete geometric analysis near of finite type, and then we construct a family of plurisubharmonic functions with maximal Hessian near . However, this construction is really technical and known only for special types of domains mentioned above, but not for arbitrary pseudoconvex domains of finite type in , even for case. Meanwhile, it is useful to understand the behavior of a holomorphic function near if we have precise estimates of the invariant metric along some curves.

In the sequel, we let be a smoothly bounded pseudoconvex domain in with smooth defining function and let . Let be Catlin’s multitype [11]. Thus, is the type in the sense of “Bloom-Graham.” If , then is an -extensible domain [12] and Herbort [10] got an estimate in this case. Here, denotes finite -type in the sense of D’Angelo. Thus, we assume that . Regular finite 1-type at is the maximum order of vanishing of for all one complex dimensional regular curves , and . We denote the regular finite 1-type at by . Note that is a positive integer and .

Assuming that , there exist coordinate functions defined in a neighborhood of such that and on for a uniform constant , and vanishes to order , and (Theorem 2.1 in [13]). With these coordinates at hand, set Then, , , and form a basis of provided is sufficiently small. For any integer , , set and define Let be a holomorphic tangent vector at and set Let be the admissible curve defined in (20). Our main result is as follows.

Theorem 1. Let be a smoothly bounded pseudoconvex domain and assume is a point of finite 1-type in the sense of D’Angelo; that is, . Then, there exist a neighborhood about , an admissible curve connecting and , and positive constants and such that, for all at ,

To prove Theorem 1, we use special coordinates constructed in Section 2 of [13]. Thus, there is a special direction , , so that, for each , the two-dimensional slice becomes a pseudoconvex domain of finite type in , whose type is less than or equal to . We then apply the method which holds for the domains of finite type in as in [6]. To avoid the difficulty to push out the domain in -direction, we use a bumping theorem of Cho [14].

2. Special Coordinates

Let and be as in Section 1. We may assume that . In this section, we consider special coordinates defined near and then construct “balls" which are of maximal size on which changes by no more than some prescribed number . In the following, we let and be multi-indices with respect to variables. In Theorem 2.1 of [13], You constructed special coordinates which represent the local geometry of near .

Theorem 2. Let be a smoothly bounded pseudoconvex domain in with smooth defining function and assume , . Then, there is a holomorphic coordinate system about such that where and where

Remark 3. (1) The second condition in (6) and the property (7) say that vanishes to order along axis and order along axis. These properties are crucial for the construction of maximal polydiscs contained in .
(2) There are much more terms (mixed with and and their conjugates) in the summation part of (6) compared to the -extensible domain cases.

According to Proposition 2.6 and Remark 2.7 of [13], there are pairs of integers , , such that the terms satisfying and with and are dominant terms in the summation part of (6). Also, there is a small constant and a fixed direction , , in direction, such that, for each fixed and for all satisfying , those major terms in the summation part of (6) satisfy where and where and .

Now, let us fix with and consider the two-dimensional slice . For each near , set , where is the projection of onto along direction. On , following the argument in two-dimensional case as in the proof of Proposition 1.1 in [6], we construct special coordinates about so that, in terms of new coordinates, there are no pure terms in variable in the expression of in (6).

Proposition 4. For each fixed , there exists a holomorphic coordinate system , , where is defined by and the function , given by , , satisfies where

In view of (6) and (8), the major terms in (10) are where for some and with and . Also, from (8), it follows that and these terms control the error terms in . As in Section 1 in [6], set and for each sufficiently small , we set Thus, for all with , it follows from (8) and (14) that and hence the summation part of (10) is dominated by .

For each near , set , where is the function defined in Proposition 4. For each small , set For each , let and define and set , where is replaced by in . The following theorem is about the existence of plurisubharmonic function with maximal Hessian. In [6], for the domains in , Catlin constructed the functions with maximal Hessian on the strip . However, for regular finite type pseudoconvex domains in , we show that the functions have maximal Hessian on each ball and this will suffice to prove the boundary behavior of the invariant metrics. The proof of the following theorem can be found in Theorem 3.2 in [9].

Theorem 5. There is a small constant such that, for each small , there is a plurisubharmonic function with the following properties:(i), ,(ii)for all at , where ,  (iii)if , where is defined in (10) for , then holds for all , where .

Let be a curve defined by for sufficiently small and . In the sequel, for each , set and set . In view of Proposition 3.4 in [9], there is a uniform small constant such that , and hence provided and are sufficiently small. In particular, we have . Note that , and for , we note that . Thus, as in Proposition 1.3 and Corollary 1.4 in [6], we obtain that where is defined in (3). In the sequel, we set , , and . If we use the plurisubharmonic weight functions constructed in Theorem 5 and follow the method to prove Theorem 6.1 in [6], we get the following estimates of the Bergman kernel along .

Theorem 6. Let be a point of regular finite 1-type and . Then, , the Bergman kernel function of at , , satisfies

3. Metric Estimates

In this section, we estimate the behavior of the invariant metric along . In [15], Hahn got the following inequalities: Therefore, the estimates for the lower bounds of will suffice for the lower bounds of and . First, we recall the following bumping theorem [14].

Theorem 7 (Theorem 2.3 in [14]). Let be a point of finite 1-type in the boundary of a pseudoconvex domain , defined by . Then, there exist and a smooth 1-parameter family of pseudoconvex domains , , each defined by , where has the following properties:(1) is smooth in for near and in for ;(2), for ;(3);(4);(5)for in , .

Proof of Theorem 1. In the sequel, let us fix and, for each , set and consider the special coordinates and , where is defined in Proposition 4. From (9), we see that . We will estimate the metrics at . For all small and for each , define where is defined in (13) with replaced by . Let be the fixed constant determined in (21). Note that . Set and, for each , define and for all small set . By (21), we see that for all . Note that the domains are pushed out only in and directions but not in direction. To avoid the difficulty to push out in direction, we use a bumping family of Theorem 7. Consider a bumping family of pseudoconvex domains with front and set . For each , let be a ball of radius with center at and set . Then, there is such that for all sufficiently small , , and .
In view of the proof in Section 3 of [13], we have and there is a uniformly (independent of ) bounded function which is holomorphic on and satisfies where . Here, we may assume that . In the sequel, we let be a vector field given by . If , then set . Otherwise, set . From (29), we note that
Let , where is the unit polydisc in , such that if , , and set and set . Then, . Since is bounded independent of (and hence independent of ), there exists a constant , independent of , such that . We want to correct so that the corrected function becomes a uniformly bounded holomorphic function on satisfying the estimate (30) with replaced by . With bumped domain at hand, set . On , instead of , we will employ weighted estimates of that is essentially a replication of the proof of Theorem 6.1 in [6].
Let be the weight function defined in Theorem 5 and set . By replacing by , we can obviously assume that is strictly plurisubharmonic on and . In view of Theorem 5, we also have From property (iii) in Theorem 5, there is a small constant , (independent of , ), so that where If we set , it follows, from (33), that
In the sequel, we set for each small . For each , set Then, is a -closed smooth -form with . Let be a smooth convex increasing function that satisfies for and for . Now, define According to the weighted estimates of -equation on (instead of ) and by using estimate (32) for each , there is which satisfies , and Since and , it follows from (38) that We consider the integrand of the last integral. If , then , so the -term in the exponent predominates. On the other hand, if and , then (35) shows that the integrand tends to zero. Thus, for any , there exist and a function so that and
Since , it follows, from the property (iii) of Theorem 5, that there is , independent of , such that and for all . Note that is independent of for , and is holomorphic in . By mean value theorem, we have and hence it follows that Now, set . Then, is holomorphic on . Since , it follows, from (30) and (42), that satisfies provided is sufficiently small.
We want to show that , where is independent of . Recall that is fixed. Thus, from the property (iii) of Theorem 5, there is a uniform constant such that . Let be the constant satisfying (28) and assume that . Since is holomorphic on , it follows, by (40) and mean value theorem, that there exists a constant , independent of , such that We need to show the boundedness of outside . Let and be smooth cutoff functions with and set , . By Kohn’s theorem on global regularity for the -equation, the following estimate for the solution of , holds on provided is sufficiently large. Note that because for all sufficiently small . Thus, we conclude from (40), (46), and the Sobolev lemma that where is independent of .
Combining (44) and (47) and by the fact that , we conclude that where is independent of and . Therefore, it follows from (43) and (48) that On the other hand, the polydisc about lies in . So one obtains that Thus, one concludes from (49) and (50) that
Set , , where ’s are defined in (1) in terms of -coordinates defined in Theorem 1. At , from the holomorphic coordinate change of in Proposition 4, we see that where and and where , . Since , , and , independent of , it follows that , . Thus, if the vector is written as , then it follows that
Let us write , and . From (51), (53), and the invariance property of the metric, it follows that
To obtain an upper bound for the Bergman metric, we note that . Thus, by elementary estimates, for any , we obtain that for . Therefore, it follows that where Combining (23) and (56), one concludes that
To estimate the upper bound of the Kobayashi metric, set Then, defines a map with . Hence, Combining (51), (58), and (61), we obtain that and hence the invariance property implies that If we combine (3), (4), (22), and (63), a proof of Theorem 1 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study was partially supported by Sogang University Research Grant of 2012.