IINVARIANCE OF DIEDERICH-FORNAESS INDEX
JIHUN YUM
Abstract.
We show that the Diederich-Fornaess index of a domain in a Stein man-ifold is invariant under CR-diffeomorphisms. For this purpose we also improve CR-extension theorem. Introduction
Let D be a relatively compact domain with C k ( k ≥
1) smooth boundary in a complexmanifold M . A C function ρ defined on a neighborhood U of D is called a (global) defining function of D if D = { z ∈ U : ρ ( z ) < } and dρ ( z ) (cid:54) = 0 for any z ∈ ∂D . In1977, K. Diederich and J. E. Fornaess ([6]) showed that, if D is a pseudoconvex domainwith C boundary and M is a Stein manifold, there exist a positive constant η with0 < η < ρ such that − ( − ρ ) η is strictly plurisubharmonicon D . The supremum of all such constant η is called the Diederich-Fornaess exponent of ρ , denoted by η ρ ( D ); if no such η exists, we define η ρ ( D ) = 0. The supremum ofall Diederich-Fornaess exponents is called the Diederich-Fornaess index of D , denotedby η ( D ). Note that a generalization to a bounded pseudoconvex domain in C n withLipschitz boundary was studied by Harrington ([11]).If D is a strongly pseudoconvex domain, then there is a strictly plurisubharmonicdefining function. Consequently, η ( D ) = 1. But the converse is not true; Fornaessand Herbig ([9], [10]) showed that, if a C ∞ smooth relatively compact domain D ⊂ C n admits a plurisubharmonic defining function, then η ( D ) = 1. In particular, the Thullendomain { ( z, w ) ∈ C : | z | + | w | < } has the Diederich-Fornaess index one, but it isclearly not strongly pseudoconvex. In fact, much more is known. For any sufficientlysmall η > η ([7]). Recently, Krantz, Liu, and Peloso ([15]) showed that for a bounded pseudoconvexdomain D ⊂ C with C ∞ smooth boundary, the Diederich-Fornaess index is one if theLevi-flat sets form a real curve transversal to the holomorphic tangent vector fields on ∂D .Many significant far-reaching conclusions follow from the condition η ( D ) >
0. Tocite only a few, we refer the reader to [2], [3], [5], [6], [12], [14], [17], and others.On the other hand, the following natural question has been posed. We express ourgratitude to M. Adachi as well as S. Yoo., for pointing out this question to us.
This research was supported by the SRC-GAIA (NRF-2011-0030044) through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education. i a r X i v : . [ m a t h . C V ] F e b i JIHUN YUM Question:
Is the Diederich-Fornaess index a biholomorphic or a CRinvariant?
Indeed, the main theorem of this article is as follows:
Theorem 1.1.
Let M , M be Stein manifolds with dimension n ( n ≥ . Let D ⊂ M , D ⊂ M be relatively compact domains with connected C k ( k ≥ smooth boundaries.If there exists a C k smooth CR-diffeomorphism f : ∂D → ∂D , then the Diederich-Fornaess indices of D and D are equal. There is a preceding result by M. Adachi ([1]) that, for a relatively compact domainin a complex manifold with C ∞ smooth Levi-flat boundary, the Diederich-Fornaess in-dex is invariant under CR-diffeomorphisms.Our proof of Theorem 1.1 requires the following modification of the theorem. Theorem 1.2.
Let M , M be Stein manifolds with dimension n ( n ≥ . Let D ⊂ M , D ⊂ M be relatively compact domains with connected C k ( k ≥ smooth boundaries. Ifthere exists C s (1 ≤ s ≤ k ) smooth CR-diffeomorphism f : ∂D → ∂D , then there existsthe unique C s smooth extension F : D → D such that F | ∂D = f and F | D : D → D is a biholomorphism. Improving CR-extension theorem
Let D be a domain in a complex manifold M , and T p ( ∂D ), T p M be the real tangentspace of ∂D , M at p ∈ ∂D , respectively. Then the complexified tangent space of T p M , C T p M := C ⊗ T p M , can be decomposed into the holomorphic tangent space T (1 , p M and the anti-holomorphic tangent space T (0 , p M , i.e. C T p M = T (1 , p M ⊕ T (0 , p M . Let T (1 , p ( ∂D ) := C T p ( ∂D ) ∩ T (1 , p M and T (0 , p ( ∂D ) := C T p ( ∂D ) ∩ T (0 , p M . Definition 2.1.
A function f ∈ C ( ∂D ) is called a CR-function on ∂D if for all p ∈ ∂D one has L ( f ) = 0 for all L ∈ T (0 , p ( ∂D ) . Definition 2.2.
Let D , D be domains in complex manifolds M , M , respectively. A C smooth map F : ∂D → ∂D is called a CR-map if F ( T (0 , p ( ∂D )) ⊂ T (0 , p ( ∂D ) for all p ∈ ∂D A C smooth map F : ∂D → ∂D is called a CR-diffeomorphism if it is a CR-map anda diffeomorphism.
Definition 2.3.
Let H p,q ( M ) be the ( p, q )-th Dolbeault cohomology of M defined by H p,q ( M ) := { ¯ ∂ -closed (p,q)-form }{ ¯ ∂ -exact (p,q)-form } NVARIANCE OF DIEDERICH-FORNAESS INDEX iii
Let H p,qc ( M ) be the ( p, q )-th Dolbeault cohomology of M with compact support definedby H p,qc ( M ) := { ¯ ∂ -closed (p,q)-form with a compact support }{ ¯ ∂ -exact (p,q)-form with a compact support } It is well known that for a bounded domain D in C n ( n ≥
2) with connected boundary,every CR-function f : ∂D → C extends to F : D → C holomorphically ([19]). Wemodify it for a relatively compact domain in a Stein manifold. Remark . The theorem above is widely known as Bochner-Hartogs theorem; but itwas actually proven by Severi and Fichera ([8], [18]; see also [20]). We are indebted toM. Range for pointing this out.
Proposition 2.5 ([16]) . Let M be a non-compact complex manifold such that H , c ( M ) =0 . For any relatively compact domain D ⊂ M with C k ( k ≥ smooth boundary suchthat M \ D is connected and any CR function f of class C s (1 ≤ s ≤ k ) on ∂D , thereis a C s smooth function F on D , holomorphic on D , such that F | ∂D = f . Lemma 2.6.
Let M be a connected smooth real manifold with dimension n ( n ≥ ,and D ⊂ M be a domain with C k ( k ≥ smooth boundary ∂D . If ∂D is connected,then M \ D is connected.Proof. Suppose that M \ D is disconnected. Then we may write M \ D as the disjointunion of two open sets X, Y ⊂ M . Then it is easy to see that ∂ ( M \ D ) = ∂X ∪ ∂Y and ∂ ( M \ D ) = ∂D . Since ∂D = ∂X ∪ ∂Y is connected by the assumption, ∂X ∩ ∂Y (cid:54) = ∅ .Let p ∈ ∂X ∩ ∂Y . Since ∂D is the C k smooth embedded submanifold of M , there is asmooth chart ψ : U p → R n at p such that • ψ ( p ) = 0, • ψ ( U p ∩ D ) = ψ ( U p ) ∩ { ( x , · · · , x n ) ∈ R n : x n < } , • ψ ( U p ∩ ∂D ) = ψ ( U p ) ∩ { ( x , · · · , x n ) ∈ R n : x n = 0 } ,where U p is an open neighborhood of p in M . Then ψ ( U p ∩ ( M \ D )) = ψ ( U p ) ∩{ ( x , · · · , x n ) ∈ R n : x n > } implies that X ∩ Y (cid:54) = ∅ , which is a contradiction. (cid:3) Theorem 2.7 (CR-extension theorem) . Let M be a Stein manifold with dimension n ( n ≥ . Let D ⊂ M be a relatively compact domain with connected C k ( k ≥ smoothboundary. If there exists a C s (1 ≤ s ≤ k ) smooth CR-function f : ∂D → C , then thereis a C s smooth function F on D , holomorphic on D , such that F | ∂D = f .Proof. Since M is a Stein manifold, M is non-compact. By Serre Duality Theorem ([4]), H , c ( M ) is dual to H n,n − ( M ) and H n,n − ( M ) = 0 since M is Stein ([13]). Therefore H , c ( M ) = 0. By Lemma 2.6, the connectedness of the boundary ∂D implies theconnectedness of M \ D . Hence the theorem follows by Proposition 2.5. (cid:3) We are now ready to prove Theorem 1.2. v JIHUN YUM
Proof of Theorem 1.2.
Since M is a Stein manifold, there is an embeddingof M into C N for some large N ∈ N , and hence from now on we may regard M asan embedded complex submanifold of C N . Therefore, we may write the given CR-diffeomorphism f as f = ( f , · · · , f N ), where each f i : ∂D → C ( i = 1 , · · · , N ) isa CR-function. By CR-extension theorem(Theorem 2.7), f extends to F : D → C N holomorphically.First, we show that the image of F is in M (i.e. F ( D ) ⊂ M ). Let p ∈ ∂D , q ∈ ∂D with f ( p ) = q . Since M is a Stein manifold, there exists a holomorphic map φ q : M → C n such that φ q is a local biholomorphism at q . Define g q : ∂D → C n by g q := φ q ◦ f .By CR-extension theorem(Theorem 2.7), g q extends to G q : D → C n holomorphically.Let U p be an open neighborhood of p in M such that φ − q is well-defined on G q ( U p ∩ D ).Define H p = f − ( q ) : U p ∩ D → M by H p := φ − q ◦ G q . Patching H p for each p ∈ ∂D , wemay construct H : U ∩ D → M , holomorphic on U ∩ D , for some neighborhood U of ∂D . Since the holomorphic extension of CR-function is unique, H is well-defined and F | U ∩ D = H . Therefore F ( U ∩ D ) ⊂ M .Now let ˜ p ∈ U ∩ D , p ∈ D and γ be a curve in D joining ˜ p and p . We choose m ∈ N and p j , q j , U j , W j , V j ( j = 1 , · · · , m ) as follows. • p := ˜ p , p m := p , and p j ∈ γ for every j . • { U j } mj =1 is an open cover of γ with p j ∈ U j , and U j ∩ U j +1 (cid:54) = ∅ for every j . • q j = F ( p j ) for every j . • W j is a neighborhood of q j in C N such that F ( U j ) ⊂ W j and V j := W j ∩ M isthe zero set of holomorphic functions ψ jα : W j → C ( α = 1 , · · · , N − n ) whenever V j is non-empty for every j .Since ( U ∩ D ) ∩ U (cid:54) = ∅ and ( ψ α ◦ F )(( U ∩ D ) ∩ U ) ≡
0, ( ψ α ◦ F )( U ) ≡ α = 1 , · · · , N − n , (i.e. F ( U ) ⊂ M ). Similarly, since ( ψ α ◦ F )( U ∩ U ) ≡ ψ α ◦ F )( U ) ≡ α = 1 , · · · , N − n , (i.e. F ( U ) ⊂ M ). By induction, it followsthat F ( U m ) ⊂ M , which means F ( p ) ∈ M . Since p ∈ D is arbitrary, F ( D ) ⊂ M . Fig. 1.
NVARIANCE OF DIEDERICH-FORNAESS INDEX v
Next, we show that F ( D ) = D . Take a C k smooth extension of F on an openneighborhood of D , and denote it again by F . Then we proceed with the followingthree steps:Step1) det ( J C F ) (cid:54) = 0 on ∂D : Choose any point p ∈ ∂D and let q := F ( p ) ∈ ∂D .Let { x , y , · · · , x n , y n } be a real coordinate system at p such that (cid:8) ∂∂x (cid:12)(cid:12) p , ∂∂y (cid:12)(cid:12) p , · · · , ∂∂x n (cid:12)(cid:12) p (cid:9) spans T p ( ∂D ). Let J and J be the complex structures of M and M , respectively.Since F is holomorphic on D , ¯ ∂F = 0 (i.e. J ◦ dF = dF ◦ J ) on D . Conse-quently, det ( J R F ) = | det ( J C F ) | holds at p . Therefore it is enough to show that d p F : T p M → T q M is R -linear isomorphism. If d p F (cid:0) ∂∂y n (cid:12)(cid:12) p (cid:1) = 0, then d p F (cid:16) − ∂∂x n (cid:12)(cid:12)(cid:12) p (cid:17) = d p F (cid:16) J (cid:16) ∂∂y n (cid:12)(cid:12)(cid:12) p (cid:17)(cid:17) = J ◦ d p F (cid:16) ∂∂y n (cid:12)(cid:12)(cid:12) p (cid:17) = 0 , which contradicts that F | ∂D is a diffeomorphism. Suppose that d p F (cid:16) ∂∂y n (cid:12)(cid:12)(cid:12) p (cid:17) = w (cid:54) =0 ∈ T q ( ∂D ). Since F | ∂D is a diffeomorphism, there exists v ∈ T p ( ∂D ) such that d p F ( v ) = w . Then d p F (cid:16) span (cid:110) ∂∂x n (cid:12)(cid:12)(cid:12) p , ∂∂y n (cid:12)(cid:12)(cid:12) p (cid:111)(cid:17) = d p F (span { v, J v } ) = span { w, J w } ⊂ T q ( ∂D ) , andspan (cid:110) ∂∂x n (cid:12)(cid:12)(cid:12) p , ∂∂y n (cid:12)(cid:12)(cid:12) p (cid:111) ∩ span { v, J v } = { } imply that there exists X ∈ span { v, J v } ⊂ T p ( ∂D ) such that d p F ( X ) = d p F (cid:0) ∂∂x n (cid:1) with X (cid:54) = ∂∂x n . This contradicts that F | ∂D is a diffeomorphism. Therefore d p F (cid:0) ∂∂y n (cid:12)(cid:12) p (cid:1) / ∈ T q ( ∂D ) and d p F : T p M → T q M is R -linear isomorphism.Step2) det ( J C F ) (cid:54) = 0 on D : Suppose that Z := { z ∈ D : det ( J C F | z ) = 0 } isnon-empty. Note that Z is a well-defined closed ( n − D . Since det ( J C F ) (cid:54) = 0 on ∂D by (1), Z is a compact analytic variety in D . Since M is Stein, Z should be finite (i.e. dim C ( Z ) = 0). This contradicts that n ≥ F ( D ) = D : Since det ( J C F ) (cid:54) = 0 on D by (2), F | D is a local biholomor-phism, hence an open map. Therefore F ( ∂D ) = ∂ ( F ( D )) holds so F ( D ) is either D or M \ D . Since M \ D is non-compact, F ( D ) = D .Now, we show that F | D : D → D is a biholomorphism. Since f − : ∂D → ∂D is also a C k smooth CR-diffeomorphism, by using the same argument as above, thereexists the holomorphic extension G of f − such that G ( D ) = D . Now consider themap G ◦ F : D → D . Since G ◦ F is holomorphic on D and G ◦ F | ∂D = id ∂D , G ◦ F = id D . Similarly, by using the same argument, F ◦ G = id D . Therefore, F | D is a biholomorphism. (cid:3) Invariance of Diederich-Fornaess index
In Theorem 1.2, we extended the given CR-diffeomorphism f : ∂D → ∂D to theinterior of D holomorphically. In this section, we extend it to the exterior of D i JIHUN YUM smoothly preserving the injectivity, and prove that the Diederich-Fornaess index isinvariant under CR-diffeomorphisms. Lemma 3.1.
Let M , M be Stein manifolds with dimension n ( n ≥ . Let D ⊂ M , D ⊂ M be relatively compact domains with connected C k ( k ≥ smooth boundaries.If there exists a C k smooth CR-diffeomorphism f : ∂D → ∂D , then there exist neigh-borhoods U ⊂ D , U ⊂ D , and a C k smooth diffeomorphism F : U → U such that F | ∂D = f and F | D : D → D is a biholomorphism.Proof. By Theorem 1.2, there exists the C k smooth extension F : D → D suchthat F | D : D → D is a biholomorphism. Take a C k smooth extension of F onan open neighborhood of D , and denote it again by F . We show that there existneighborhoods U ⊂ D , U ⊂ D such that F | U : U → U is a diffeomorphism. Inthe proof of Theorem 1.2, we showed that det ( J R F | p ) (cid:54) = 0 for all p ∈ ∂D . Hence F isa local diffeomorphism at every p ∈ ∂D .Next, choose any Riemannian metric g on M . Define dist( w ) : M → R by thedistance from w to ∂D with respect to g . Let V δ := { w ∈ M : dist( w ) < δ } . Choose δ > w ∈ V δ there exists the unique closed point in ∂D from w .Define ψ ( w ) : V δ → ∂D by the closed point in ∂D from w . For each q ∈ ∂D , define V q := { w ∈ V δ : ψ ( w ) ∈ ∂D ∩ B ( q, (cid:15) ) } , where B ( q, (cid:15) ) is the geodesic ball centered at q with radius (cid:15) >
0. Let U F − ( q ) be the connected component of F − ( V q ) containing F − ( q ). By letting (cid:15) > , δ > F | U p : U p → V F ( p ) is injective forall p ∈ ∂D . Let U := (cid:83) p ∈ ∂D U p and V := (cid:83) q ∈ ∂D V q , then U and V are neighborhoods of ∂D and ∂D , respectively.Now choose any z , z ∈ U \ D . Then z ∈ U p , z ∈ U p for some p , p ∈ ∂D . Let q := F ( p ) , q := F ( p ). Suppose that w := F ( z ) = F ( z ) ∈ V q ∩ V q . If p = p ,then since F | U p : U p → V q is a diffeomorphism, z = z . If p (cid:54) = p , then the distancerealizing geodesic γ joining w and q := ψ ( w ) is in V q ∩ V q . Note that ( F | U p ) − ( γ ),( F | U p ) − ( γ ) are curves joining z , F − ( q ) and z , F − ( q ), respectively. Therefore if z (cid:54) = z , then F | U p : U p → V q is not injective near F − ( q ), which is a contradiction.By letting U := U ∪ D , U := V ∪ D , Lemma 3.1 is proved. (cid:3) Fig. 2.
NVARIANCE OF DIEDERICH-FORNAESS INDEX vii
Proof of Theorem 1.1.
By Lamma 3.1, there exists a C k smooth diffeomor-phism F : U → U such that F | ∂D = f and F | D : D → D is a biholomorphism,where U and U are neighborhoods of D and D , respectively. If η ( D ) and η ( D )are both zero, then we are done. So we may assume that η ( D ) >
0. Let ρ be a C s (1 ≤ s ≤ k ) smooth defining function of D , then ρ := ρ ◦ F is a C s smooth definingfunction of D . If − ( − ρ ) η is strictly plurisubharmonic on D for some 0 < η ≤
1, then − ( − ( ρ ◦ F )) η is also strictly plurisubharmonic on D with the same η because F isholomorphic on D . This implies that η ( D ) ≥ η , and hence η ( D ) ≥ η ( D ). Therefore η ( D ) >
0. Applying the same argument as above with F − : U → U , one can seethat η ( D ) ≤ η ( D ). Hence η ( D ) = η ( D ). (cid:3) Acknowledgements : The author would like to express his deep gratitude to ProfessorKang-Tae Kim for valuable guidance and encouragements, and to Professor R. M.Range for valuable advice. This work is part of author’s Ph.D. dissertation at PohangUniversity of Science and Technology.
References
1. M. Adachi,
A CR proof for a global estimate of the Diederich-Fornaess index of Levi-flat realhypersurfaces , Complex analysis and geometry, 41-48, Springer Proc. Math. Stat., , Springer,Tokyo. (2015)2. B. Berndtsson and P. Charpentier,
A Sobolev mapping property of the Bergman kernel , Math. Z. , no. 1, 1-10. (2000)3. Z. B(cid:32)locki,
The Bergman metric and the pluricomplex Green function , Trans. Amer. Math. Soc. , no. 7, 2613-2625. (2005)4. D. Chakrabarti and M.-C. Shaw, L Serre Duality on domains in complex manifolds and applica-tions , Trans. Amer. Math. Soc. , no. 7, 3529-3554. (2012)5. B.-Y. Chen and S. Fu,
Comparison of the Bergman and Szeg¨o kernels , Adv. Math. , no. 4,2366-2384. (2011)6. K. Diederich and J. Fornæss,
Pseudoconvex Domains: bounded strictly plurisubharmonic exhaus-tion functions , Invent. Math. , no. 2, 129-141. (1977)7. K. Diederich and J. Fornæss, Pseudoconvex Domains: an example with nontrivial neighborhood ,Math. Ann. , no. 3, 275-292. (1997)8. G. Fichera,
Caratterizzazione della traccia, sulla frontiera di un campo, di una funzione analiticadi pi ` u variabili complesse , Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) A note on plurisubharmonic defining functions in C , Math. Z. , no. 4, 769-781. (2007)10. J. E. Fornæss and A. K. Herbig, A note on plurisubharmonic defining functions in C n , Math. Ann. , no. 4, 749-772. (2008)11. P. S. Harrington, The order of plurisubharmonicity on pseudoconvex domains with Lipschitz bound-aries , Math. Res. Lett. , no. 3, 485-490. (2008)12. P. S. Harrington, Global regularity for the ¯ ∂ -Neumann operator and bounded plurisubharmonicexhaustion functions , Adv. Math. , no. 4, 2522-2551. (2011)13. L. H¨ormander, An introduction to complex analysis in several variables , Third edition. North-Holland Mathematical Library, . North-Holland Publishing Co., Amsterdam. (1990)14. J. J. Kohn, Quantitative estimates for global regularity , Analysis and geometry in several complexvariables (Katata, 1997), 97-128, Trends Math., Birkhuser Boston, Boston, MA. (1999) iii JIHUN YUM
15. S. G. Krantz, B. Liu and M. Peloso,
Geometric Analysis on the Diederich-Fornæss Index ,arXiv:1606.02343. (2016)16. C. Laurent-Thiebaut,
Holomorphic function theory in several variables , Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, xiv+252 pp. (2011)17. S. Pinton and G. Zampieri,
The Diederich-Fornaess index and the global regularity of the ¯ ∂ -Neumann problem , Math. Z. , no. 1-2, 93-113. (2014)18. F. Severi, Risoluzione generale del problema di Dirichlet per le funzioni biarmoniche , Rend. RealeAccad. Lincei Holomorphic functions and integral representations in several complex variables ,Graduate Texts in Mathematics, , Springer-Verlag, New York, (1986)20. R. M. Range,
Extension phenomena in multidimensional complex analysis: correction of the his-torical record , Math. Intelligencer , no. 2, 4-12. (2002) Department of Mathematics, Pohang University of Science and Technology, Po-hang, 790-784, Republic of Korea
E-mail address ::