A Pseudoconvex Domain? R

WHATIS... ? a Pseudoconvex Domain? R. Michael Range Pseudoconvexity is a most central concept in mod- on D, but surely it has no holomorphic extension ern complex analysis. However, if your training to any neighborhood of P.2 in that area is limited to functions of just one Hartogs’s example is so amazing and histori- complex variable, you probably have never heard cally significant, and yet completely elementary, of it, since every open subset of the complex that it deserves to be presented in any ex- plane C is pseudoconvex. Pseudoconvexity or, bet- position of the subject. Consider the domain H ⊂ C2 = {(z, w) : z, w ∈ C} defined by ter, nonpseudoconvexity, is a higher-dimensional phenomenon. Incidentally, this is true also for H = {(z, w) : |z| < 1, 1/2 < |w| < 1} Euclidean convexity: every open connected subset ∪ {|z| < 1/2, |w| < 1}. of R is convex. Life is definitely more interesting Let f : H → C be holomorphic and fix r with in higher dimensions! 1/2 < r < 1. The function Pseudoconvexity is so central because it relates 1 f (z, ζ) dζ to the very core of holomorphic (i.e., complex an- F(z, w) = 2πi Z = ζ − w alytic) functions, which is intimately intertwined |ζ| r with power series, the identity theorem, and an- is easily seen to be holomorphic on G = {(z, w) : alytic continuation. This concept has its roots in |z| < 1, |w| < r}. Observe that for fixed z0 with | | → Friedrich Hartogs’s surprising discovery in 1906 z0 < 1/2 the function w f (z0, w) is holomorphic on the disc {|w| < 1}, and hence, by of a simple domain H in C2 with the property that the Cauchy integral formula, f (z , w) = F(z , w) every function that is holomorphic1 on H has a 0 0 for |w| < r. Thus f ≡ F on {(z, w) : |z| < 1/2, holomorphic extension to a strictly larger open set |w| < r}, which implies f ≡ F on H ∩G by the iden- H. In dimension one there is no such thing! In fact, tity theorem, so that F provides the holomorphic ⊂ C ifb P is a boundary point of a domain D , the extension of f from H to H = H ∪ G. function fP (z) = 1/(z − P) is clearly holomorphic Hartogs’s discovery immediately raises the b fundamental problem of characterizing those domains D ⊂ Cn for which holomorphic exten- R. Michael Range is professor of mathematics at the State sion of all holomorphic functions on D does NOT University of New York at Albany. His email address is hold. Such domains are called domains of holo- [email protected]. morphy. More precisely, according to this point DOI: http://dx.doi.org/10.1090/noti798 of view, D is a domain of holomorphy, if for 1Holomorphic functions in several complex variables are each boundary point P ∈ bD there exists a func- n those continuous functions on an open set D ⊂ C which tion fP holomorphic on D which does not extend are holomorphic in each variable separately. Iteration of the Cauchy integral formula for discs readily implies 2This sort of simple construction does not extend to more that such functions are then C∞ in all underlying real than one variable, as the zeroes and singularities of holo- variables and have local power series expansions in the morphic functions are not isolated in the case of two or complex variables. more variables. The reader may find more details in [3]. February 2012 Notices of the AMS 301 holomorphically to any neighborhood of P.3 As Note that if D ⊂ C, the restriction on t is mentioned earlier, every domain in the complex satisfied only for t = 0, so the conditions in A) and plane is trivially a domain of holomorphy, and B) trivially hold in this case. this easily implies that every product domain Levi’s results made it clear that the “complex D = D1 × D2 × · · · × Dn with Dj ⊂ C, j = 1, . , n, Hessian” LP (r; t)—now universally called the Levi is also a domain of holomorphy. Furthermore, it form—plays a fundamental role in the charac- is elementary to show that every Euclidean convex terization of domains of holomorphy. The term domain D ⊂ Cn is a domain of holomorphy. Har- “pseudoconvex” was introduced in this context in togs, of course, produced the first example of a the influential 1934 “Ergebnisbericht” of H. Behnke domain which is NOT a domain of holomorphy. and P. Thullen, which summarized the status and The reader may consult [2] for other surprising principalopenquestions inmultidimensionalcom- consequences of Hartogs’s discovery. plex analysis at that time. To distinguish Levi’s The essenceof pseudoconvexity is now captured differential conditions from other formulations of by the following statement. pseudoconvexity, one refers to the condition in Pseudoconvexity is that local analytic/geo- A) as Levi pseudoconvexity. If the stronger version metric property of the boundary of a domain D in B) holds, one says that D is strictly or strongly in Cn which characterizes domains of holomor- pseudoconvex at P. phy. By Levi’s result, if D is strictly pseudoconvex at Note that it is not at all clear that the global every boundary point, then D is locally a domain property of being a domain of holomorphy should of holomorphy. The emphasis on “locally” is criti- allow a purely local characterization, i.e., some- cal. Levi himself recognized that his result was far thing that can be recognized by just looking near from yielding the wished-for corresponding global each boundary point. In fact, the validation of this version. For many years it remained a central open statement was the culmination of major efforts problem—known as the Levi problem—to show over a period of more than forty years. that a strictly pseudoconvex domain is indeed Just a few years after Hartogs’s discovery, a domain of holomorphy. Solutions were finally E. E. Levi studied domains of holomorphy with obtained in the early 1950s by K. Oka, H. Bre- differentiable boundaries. He found the following mermann, and F. Norguet, thereby vindicating the simple differential condition, which is remarkably central role of pseudoconvexity. similar to the familiar differential characterization The extension to arbitrary domains requires an of Euclidean convexity. We assume that D ∩ U = appropriate definition of pseudoconvexity. Many {z ∈ U : r(z) < 0}, where r is a C2 real-valued equivalent versions have been introduced over the function with dr ≠ 0 on a neighborhood U of years. Perhaps most elegant is a formulation that P ∈ bD. involves the notion of plurisubharmonic function introduced by Oka and P. Lelong in the 1940s.5 Theorem (Levi, 1910/11). A) If there exists a holo- Suffice it to say that a C2 function r on D is morphic function on D ∩ U which does not extend plurisubharmonic precisely when its Levi form holomorphically to P (in particular, if D is a domain satisfies L (r; t) ≥ 0 for all t ∈ Cn and z ∈ D of holomorphy), then z and that general plurisubharmonic functions can 2 n ∂2r be well approximated from above by C or even n ∞ LP (r; t) = (P) tj tk ≥ 0 for all t ∈ C C plurisubharmonic functions. Let us denote by ∂z ∂z j,kX=1 j k dist(z, bD) the Euclidean distance from z to bD. n ∂r Definition. A domain D ⊂ Cn is said to be with (P) tj = 0. jX=1 ∂zj pseudoconvex (or Hartogs pseudoconvex) if the function ϕ(z) = − log dist(z, bD) is plurisubhar- B) If LP (r; t) > 0 for all t ≠ 0 which satisfy n ∂r monic on D. (P) tj = 0, then the neighborhood U can be j=1 ∂zj chosenP so that U ∩ D is a domain of holomorphy.4 Note that ϕ is a continuous function which tends to ∞ as z → bD. One verifies that convex 3This definition is formally weaker than the one com- domains are pseudoconvex and that a domain with 2 monly found in the literature; namely, a domain of holo- C boundary is pseudoconvex according to this morphy is a domain on which there exists a single holo- definition if and only if it is Levi pseudoconvex. morphic function which cannot be extended holomorphi- Also, any pseudoconvex domain is the increasing cally to any of its boundary points. However, by a 1932 fundamental theorem of H. Cartan and P. Thullen, the 5Subharmonic functions were first introduced in one two notions are in fact equivalent. complex variable by F. Hartogs in 1906. That concept 4 With zj = xj + i yj , j = 1, . , n, the complex par- was generalized in the obvious way to n real variables in tial differential operators ∂/∂zj are defined by ∂/∂zj = the 1920s. In contrast, plurisubharmonic functions are (1/2)(∂/∂xj − i ∂/∂yj ); an analogous definition holds for those functions of n complex variables which are sub- their conjugates ∂/∂zj . harmonic on each complex line where defined. 302 Notices of the AMS Volume 59, Number 2 union of strictly (Levi) pseudoconvex domains strict convexity. Unfortunately, this neat charac- with C∞ boundaries. Incidentally, it is known that terization breaks down already in the case of a domain D is (Euclidean) convex if and only if simple pseudoconvex domains of finite type, as − log dist(z, bD) is a convex function. shown by an example discovered by J. J. Kohn The general version of the solution of Levi’s and L. Nirenberg in 1972. However, if one drops problem is then stated as: all regularity conditions of the coordinates on the A domain in Cn is a domain of holomorphy if boundary, one is left with the following tantalizing 6 and only if it is pseudoconvex.

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