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1 Math. Proc. Camb. Phil. Soc. 2014), 156, c Cambridge Philosophical Society 2014 doi: /s First published online 19 February On automorphism groups of generalized Hua domains BY FENG RONG Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, , P.R. China. frong@sjtu.edu.cn Received 3 June 2013; revised 11 November 2013) Abstract Hua domains, generalized Hua domains Hua constructions, named after the great Chinese mathematician Luogeng Hua Loo-Keng Hua), are generalizations of Cartan Hartogs domains introduced by Weiping Yin around the end of the 20th century. In this paper, we give a complete description of automorphism groups of generalized Hua domains. We also discuss the corresponding problem for Hua constructions. 1. Introduction Let C N be a bounded domain. The automorphisms group Aut ) of is the collection of one-to-one onto holomorphic mappings ϕ : whose inverse is automatically holomorphic, cf. [10]), which forms a group under the binary operation of composition of mappings. It follows from the work of H. Cartan that Aut ) is a finite dimensional real Lie group cf. [9]). The study of automorphism groups of complex domains is a classical topic in complex analysis. It is a particularly powerful tool in several complex variables. For example, using automorphism groups, Poincaré showed that balls polydiscs in multi-dimensions are not biholomorphically equivalent, hence there is no Riemann Mapping Theorem in several complex variables. From the work of Bedford Dadok [2] Saerens Zame [13], it is known that every compact Lie group can be realized as the automorphism group of a strictly pseudoconvex bounded domain with real analytic boundary in C N. On the other h, from the work of B. Wong [15], a C strongly pseudoconvex domain with a noncompact automorphism group is necessarily biholomorphic to the unit ball. An outsting problem concerning noncompact automorphism groups is the Greene-Krantz conjecture cf. [5, 6]): A boundary orbit accumulation point of a smoothly bounded domain in C N is a point of finite type. By Bedford Pinchuk [3], if the automorphism group of a bounded domain with real analytic pseudoconvex boundary in C 2 is noncompact, then it is biholomorphic to the domain {w, z) C 2 : w 2μ + z 2 < 1, μ Z + }. Domains of the form {w, z) C C n : w 2μ + z 2 < 1} with μ>0 are called egg The author is partially supported by the National Natural Science Foundation of China Grant No ).

2 462 FENG RONG domains. Rewriting w 2μ + z 2 < 1as w 2μ < 1 z 2 noticing that 1 z 2 > 0 defines the unit ball, one can regard egg domains as domains built upon the unit ball which is a very special bounded symmetric domain. Thus one can try to construct domains which are built upon other bounded symmetric domains. Let D C n be a bounded symmetric domain. Denote by N D z,w)its generic norm with N D z, z) >0 for all z D see e.g [4, 14]). Then one can define Cartan Hartogs domains as {w, z) C m D : w 2μ < N D z, z), μ > 0}. A complete description of the automorphism groups of such domains are given in [1]. Set m = m m r write w = w 1,...,w r ) C m 1... Cm r. As the generalization of Cartan Hartogs domains, one can define Hua domains as cf. [16]) w 1,...,w r, z) C m 1 C m r D : w j 2μ j < N D z, z), μ j > 0. More generally, one can define generalized Hua domains as w 1,...,w r, z) C m 1 C m r D : w j 2μ j < N D z, z) ν,μ j,ν >0. A domain C N is called homogeneous if its automorphism group is transitive on. In Section 2, we will show that the only homogeneous generalized Hua domain is the unit ball i.e. with D being the unit ball each μ j = 1 ν = 1). Our main result is the following complete description of the automorphism groups of inhomogeneous generalized Hua domains. THEOREM 1 1. Let H D be an inhomogeneous generalized Hua domain i.e. not the unit ball). Then its automorphism group AutH D ) consists entirely of the following mappings w, z) = U 1 w 1 )φ 1 z),...,u r w r )φ r z), ϕz)), 1 1) where ϕz) AutD),U j w j ) Um j ) the unitary group of degree m j ) with z 0 = ϕ 1 0). φ j z) = N Dz 0, z 0 ) ν/2μ j N D z, z 0 ) ν/μ j, In Section 2, we recall some basic definitions results. In Section 3, we prove Theorem 1 1. In Section 4, we discuss Hua constructions, which are further generalization of Hua domains. 2. Preliminaries A bounded domain D C n is called a bounded symmetric domain if at each z D there exists a biholomorphism σ z : D D such that σ 2 = id z is an isolated fixed point of σ z. From the work of É Cartan, irreducible bounded symmetric domains in C n can be divided into four types of classical domains two exceptional types cf. [4, 7, 8]). We list below the four classical bounded symmetric domains their generic norms.

3 On automorphism groups of generalized Hua domains 463 Type I domains I p,q 1 p q): V is the space of p q complex matrices, D ={z V : I z z t > 0}, N D z,w)= deti z w t ). Type II domains II n : V is the space of skew symmetric n n complex matrices, D ={z V : I + z z > 0}, N D z,w) 2 = deti + z w). Type III domains III n : V is the space of symmetric n n complex matrices, D ={z V : I z z > 0}, N D z,w)= deti z w). Type IV domains IV n : Denote Qz,w)= n i=1 z iw i, D ={z C n : 1 2Qz, z) + Qz, z) 2 > 0, Qz, z) <1}, N D z,w)= 1 2Qz, w) + Qz, z)q w, w). For simplicity, we will write Nz, z) for N D z, z). For any ϕ AutD) z,w D,wehavecf.[4, 14]) which implies Nϕz), ϕz))nϕw), ϕw)) Nϕz), ϕw)) 2 = Nz, z)nw, w) Nz,w) 2, Nϕz), ϕz)) = Nz, z)nz 0, z 0 ), 2 1) Nz, z 0 ) 2 with z 0 = ϕ 1 0), since Nz, 0) = 1 for all z D. We also have Nϕz), ϕw)) γ = Jϕz)Nz,w) γ Jϕw), 2 2) where γ is the genus of D Jϕz) is the Jacobian determinant of ϕ. A bounded domain C N is called a circular domain if it is invariant under the action of the circle group S 1 given by z 1,...,z N ) e iθ z 1,...,e iθ z N ). A circular domain is said to be complete if it contains the origin. We will need the following theorem by H. Cartan. THEOREM 2 1 H. Cartan, cf. [11]). Let C N be a bounded complete circular domain. Then, the isotropy group at the origin consists entirely of linear transformations. Let C N be a bounded domain. If its boundary is a regularly embedded C k - hypersurface k 2), then there exists a C k smooth function ρ : C N R such that ={z C N : ρz) <0} ρp) 0 for any p. The Levi form of ρ is defined as N ρ 2 ρ := dz i d z j. z i z j i, Let p.acomplex tangent vector to at p is given by t = t 1,...,t N ) C N such that N ρ p)t i = 0. z i=1 i We write t T p ). We say that is weakly) Levi pseudoconvex at p if N i, 2 ρ z i z j p)t i t j 0, t T p ).

4 464 FENG RONG If the above inequality is strict whenever t 0, then p is said to be strongly pseudoconvex. Let C N be a bounded domain p. We say that p is a boundary orbit accumulation point if there are a point q automorphisms ϕ i Aut ) such that ϕ i q) p as i. We will also need the following Wong Rosay theorem. THEOREM 2 2 Wong [15], Rosay [12]). Let C N be a bounded domain. If there exists a boundary orbit accumulation point p such that is C 2 smooth strongly pseudoconvex near p, then is biholomorphic to the unit ball in C N. 3. Automorphism groups Let H D be a generalized Hua domain which is not the unit ball. The idea of the proof of Theorem 1 1 is similar to [1]. However, the details, especially for showing pseudoconvexity in Lemma 3 2, are much more involved. LEMMA 3 1. The mappings 1 1) in Theorem 1 1 form a subgroup of AutH D ). Proof. For mappings 1 1), we have for each 1 j r U j w j )φ j z) 2 w j 2 = Nz 0, z 0 ) ν/μj Nz, z 0 ) 2ν/μ j = Nϕz), ϕz))ν/μ j Nz, z) ν/μ j where we used 2 1) for the second equality. Thus we have U j w j )φ j w, z) 2μ j = w j 2μ j, 1 j r. 3 1) Nϕz), ϕz)) ν Nz, z) ν By the definition of H D, this shows that mappings 1 1) are indeed automorphisms of H D. Let w, z) = U 1 w 1 )φ 1 z),...,u r w r )φ r z), ϕz)) w, z) = Ũ 1 w 1 ) φ 1 z),...,ũ r w r ) φ r z), ϕz)) be two mappings as in 1 1). Then w, z) = Ũ 1 U 1 w 1 )) φ 1 ϕz))φ 1 z),...,ũ r U r w r )) φ r ϕz))φ r z), ϕϕz))). Using 2 2), we have, φ j ϕz)) = N ϕ 1 0), ϕ 1 0)) ν/2μ j Nϕz), ϕ 1 0)) ν/μ j φ j z) = Nφ 1 0), φ 1 0)) ν/2μ j Nz,φ 1 0)) ν/μ j = Nϕ 1 ϕ 1 0)), ϕ 1 ϕ 1 0))) ν/2μ j Nz,ϕ 1 ϕ 1 0))) ν/μ j Jϕϕ 1 ϕ 1 0))) ν/2γμ j = Jϕz) ν/γ μ j Jϕϕ 1 ϕ 1 0))) ν/2γμ j Jϕ 1 0) ν/2γμ j Jϕ 1 ϕz)) ν/γ μ j Jϕ 1 0) ν/2γμ j. Note that Jϕz)Jϕ 1 ϕz)) = 1 Jϕϕ 1 ϕ 1 0))) ν/2γμ j Jϕ 1 0) ν/2γμ j Jϕϕ 1 ϕ 1 0))) ν/2γμ j Jϕ 1 0) ν/2γμ j = 1. Thus w, z) also has the form 1 1), which proves the lemma.

5 On automorphism groups of generalized Hua domains 465 The boundary of H D can be decomposed as where 0 H D ={0} D with H D = 0 H D 1 H D, 1 H D ={w, z) C m D : ρw,z) = 0}, ρw,z) := w j 2μ j Nz, z) ν. LEMMA 3 2. Every point of 1 H D is strongly pseudoconvex. Proof. We consider each type of classical domains separately. Since the computation for type II type III domains are similar to type I domains, we only give details for type I domains type IV domains. Type I: Let z ={z ik },1 i p, 1 k q, be the complex coordinates for a type I domain D. Since D is homogeneous the automorphisms in 1 1) extend smoothly to the boundary, it suffices to show that 1 H D is strongly pseudoconvex at a boundary point Z 0 = w, z 0 ) = w, z 11, 0,...,0). For type I domains Nz, z) = deti z z t ) = σ S p sgnσ )a 1σ1) a pσp), where S p is the symmetric group on p symbols, sgnσ ) is the signature of σ S p a rs = δ rs q t=1 z rt z st,1 r, s p, with δ rs being the Kronecker symbol. Note that only a is contains z ik, with a is / z ik = z sk, only a ri contains z ik, with a ri / z ik = z rk. Hence, we have Nz, z) z ik = σ S p sgnσ )a 1σ1) a iσi) z ik a pσp), Nz, z) z ik = σ S p sgnσ )a 1σ1) a σ 1 i)i z ik a pσp) 2 Nz, z) z ik z ik = σ S p,σ i) i + σ S p,σ i)=i sgnσ )a 1σ1) a iσi) z ik sgnσ )a 1σ1) a σ 1 i)i z ik 2 a ii z ik z ik a pσp). a pσp) At Z 0,wehaveNz, z) = 1 z 11 2 Then one readily checks that a 11 = 1 z 11 2, a ss = 1, 2 s n, a rs = 0, r s. Nz, z) Nz, z) Z 0 ) Z 0 ) = z ik z ik z 11 2 i = k = 1, 0 otherwise,

6 466 FENG RONG 2 Nz, z) z ik z ik Z 0 ) = 1 + z 11 2 i 1, k 1, 1 otherwise. For 1 j r, write w j = w j,1,...,w j,m j ). Then the Levi form of ρ at Z 0 is given by ρz 0 ) = m μ 2 j w j j 2μ j 1) dw j,l d w j,l + ν1 z 11 2 ) ν 2 1 ν z 11 2 )dz 11 d z z 11 2 ) p + 1 z 11 2 ) dz i1 d z i1 + i=2 p i=2 q dz ik d z ik k=2 )) q dz 1k d z 1k. 3 2) For 1 j r, write s j = s j,1,...,s j,m j ).Fort = s 1,...,s r, t 1,...,t pq ) T Z0 H D,it satisfies m j μ j w j 2μ j 1) w j,l s j,l + ν z 11 1 z 11 2 ) ν 1 t 1 = ) By the Levi form 3 2), we need to show that for t = s 1,...,s r, t 1, 0,...,0) 0 satisfying 3 3) we have m μ 2 j w j j 2μ j 1) s j,l 2 + ν1 z 11 2 ) ν 2 1 ν z 11 2 ) t 1 2 > ) From 3 3) we have r t 1 2 μ j w j 2μ j 1) m j w 2 j,ls j,l =. 3 5) ν 2 z z 11 2 ) 2ν 1) Note that at the boundary point Z 0 we have w j 2μ j = 1 z 11 2 ) ν, 3 6) with at least one w j 0. Using 3 5) 3 6), we can rewrite the left-h side of 3 4) as A + B ν z z 11 2 ), ν where m A = ν z 11 2 w j 2μ j μ 2 j w j j 2μ j 1) s j,l 2 k=2 2 m j B = 1 ν z 11 2 ) μ j w j 2μ j 1) w j,l s j,l.

7 On automorphism groups of generalized Hua domains 467 Thus to show 3 4), we need to show that m w j 2μ j μ 2 j w j m j j 2μ j 1) s j,l 2 μ j w j 2μ j 1) w j,l s j,l 2 > ) The left-h side of 3 7) can be rewritten as C + D with m j C = μ 2 j w m j 4μ j 1) w j 2 s j,l 2 j 2 w j,l s j,l D = 1 i j r mi w i 2μ i 1) w j 2μ j 1) m j μ 2 j w i 2 s j,l 2 ) m j )) μ i w i,k s i,k μ j w j,l s j,l. For C, we get, by the Cauchy Schwartz inequality, m j C > μ 2 j w m j 4μ j 1) w j,l 2 s j,l 2 j For D, wehave D = 1 i< j r mi w i 2μ i 1) w j 2μ j 1) mi 2 w j,l s j,l > 0. μ 2 i w j 2 s i,k 2 + μ 2 j w i 2 s j,l 2 ) m j ) μ i w i,k s i,k μ j w j,l s j,l 1 i< j r mi m j ) m j )) μ i w i,k s i,k μ j w j,l s j,l m i w i 2μ i 1) w j 2μ j 1) μ 2 i w j,l 2 s i,k 2 + μ 2 j w i,k 2 s j,l 2 μ i μ j w i,k w j,l s i,k s j,l μ i μ j w i,k w j,l s i,k s j,l ) = m i m j w i 2μ i 1) w j 2μ j 1) μ i w j,l s i,k μ j w i,k s j,l i< j r m j Therefore 3 7), hence 3 4), is valid we have shown that 1 H D is strongly pseudoconvex at Z 0. Type IV: Let z ={z 1,...,z n } be the complex coordinates for a type IV domain D. As above, it suffices to show that 1 H D is strongly pseudoconvex at a boundary point Z 0 = w, z 1, 0,...,0). For type IV domains n n ) n ) Nz, z) = 1 2 z i 2 +, i=1 i=1 z 2 i z 2 k which is equal to 1 z 1 2 ) 2 at z 1, 0,...,0). One readily checks that the Levi form of ρ at

8 468 FENG RONG Z 0 is given by m ρz 0 ) = μ 2 j w j j 2μ j 1) dw j,l d w j,l + 2ν1 z 1 2 ) 1 2ν 1) 2ν z 1 2 )dz 1 d z 1 + For t = s 1,...,s r, t 1,...,t n ) T Z0 H D, it satisfies ) n dz i d z i. 3 8) i=2 m j μ j w j 2μ j 1) w j,l s j,l + 2ν z 1 1 z 1 2 ) 2ν 1 t 1 = ) By the Levi form 3 8), we need to show that for t = s 1,...,s r, t 1, 0,...,0) 0 satisfying 3 9) we have m μ 2 j w j j 2μ j 1) s j,l 2 + 2ν1 z 1 2 ) 2ν 1) 1 2ν z 1 2 ) t 1 2 > ) Note that if we replace 2ν by ν z 1 by z 11 in 3 9) 3 10), then we get 3 3) 3 4). Therefore, arguing exactly as above we can show that 3 10) is valid. Hence, 1 H D is strongly pseudoconvex at Z 0. Remark 3 3. The above lemma does not hold for Hua constructions see Section 4 for details). LEMMA 3 4. H D is not a homogeneous domain. Proof. Assume that H D is a homogeneous domain. Then H D is a bounded symmetric domain since it is symmetric with respect to the origin. By Lemma 3 2, each point in 1 H D is strongly pseudoconvex, thus H D is of rank one, hence is the unit ball, a contradiction. In particular this implies that D is of rank one. Therefore, H D takes the form ) ν n w 1,...,w r, z) C m 1 C m r C n : w j 2μ j < 1 z i 2,μ j,ν >0. It is easy to see that the above domain is the unit ball if only if μ j = 1 for each j ν = 1. LEMMA 3 5. The set {0} D is invariant under every mapping in AutH D ). Proof. Let AutH D ) 0, z i ) be a sequence of points in {0} D converging to {0} D. Since D is homogeneous, we can choose a sequence of mappings i as in 1 1) such that i 0, 0) = 0, z i ). For any convergent subsequence of 0, z i ) = i 0, 0),if the limit is in 1 H D, then by Wong Rosay Theorem Lemma 3 2, H D is the unit ball, a contradiction by Lemma 3 4. Hence the sequence 0, z i ) can only have limit in {0} D. Write = 1, 2 ) C m D. By the above discussion, the mapping 1 0, z) vanishes identically on D, hence 1 0, z) 0onD. Thus, maps {0} D into itself. i=1

9 On automorphism groups of generalized Hua domains 469 LEMMA 3 6. If AutH D ) satisfies 0, 0) = 0, 0), then is linear takes the form w 1,...,w r, z) = A 1 w 1 ),...,A r w r ), Lz)), 3 11) where Lz) AutD) is linear with L0) = 0 A j w j ) Um j ). Proof. Since H D is a bounded complete circular domain, is linear by Cartan s Theorem. By Lemma 3 5, 0, z) = 0, Lz)) for some Lz) AutD) linear with L0) = 0. Choose a mapping as in 1 1) such that 0, Lz)) = 0, z). Then 0, z) = 0, z), hence also a linear mapping. Write w,z) = [ A 0 BI ][ ] w. z Since AutH D ), the mapping w Aw is an automorphism of the domain m w j 2μ j < 1. By definition, μ j s are mutually distinct. An easy computation shows that A must be of the form DiagA 1,...,A r ) with A j Um j ). It remains to be seen that B = 0. Choose a mapping 1 as in 1 1) such that 1 w, z) = A 1 w, z). Then For all k 1, we then have 1 w,z) = 1 ) k w, z) = [ I 0 BI [ I 0 kb I ][ ] w. z ][ ] w. z Since the isotropy group at the origin is compact, we must have B = 0. Hence = it takes the form 3 11) as claimed. The proof of Theorem 1 1 now follows easily. Proof of Theorem 1 1 Let AutH D ). By Lemma 3 5, 0, 0) = 0, z 0 ) for some z 0 D. Choose a mapping as in 1 1) such that 0, z 0 ) = 0, 0). Then AutH D ) 0, 0) = 0, 0). Thus by Lemma 3 6, takes the form 3 11), which is a particular form of 1 1). By Lemma 3 1, the mappings in 1 1) form a subgroup of AutH D ). Therefore also has the form 1 1), which completes the proof. Remark 3 7. As in [1], we can also consider the dimension of AutH D ). Comparing Theorem 1 1 [1, theorem 1 1], we see that if CH D denotes the Cartan Hartogs domain over D then dim R AutH D ) = dim R AutCH D ) m 2. m 2 i

10 470 FENG RONG Hence as in [1, proposition 4 1], we have dim R AutH D ) dim C H D )) 2 p 2 1)q 2 1) 2pqm 1) m 2 ) r m2 i n2n 1) 1 2 = nn 1)) 2 mnn 1) m 2 r n2n + 1) 1 nn + 1)) 2 2 mnn + 1) m 2 r 1 n + 2)n + 1) 2 n2 2mn m 2 ) r m2 i m2 i m2 i D = I p,q, ) D = II n, ) D = III n, D = IV n. It is easy to see that for a fixed m, m 2 r m2 i is minimal when r = 2 m 1 = 1, which gives m 2 r m2 i = 2m 1). Thus the maximal value of dim R AutH D ) dim C H D )) 2 for a genuine generalized Hua domain H D is 4 which is reached when D = I 1,1 m = r = 2. Remark 3 8. Let R r be the set defined by ={x = x 1,...,x r ) R r : 0 x i < 1, 0 x 1 + +x r < 1}. Define a map π from H D to as follows: πw,z) = w1 2μ 1 Nz, z) ν,..., w r 2μr Nz, z) ν ). 3 12) Then by the definition of H D, the map π is surjective. For each x,by3 1) Theorem 1 1, the fiber π 1 x) is invariant under AutH D ), whose action is transitive. Thus H D is a fiber-wise homogeneous domain. 4. On Hua constructions As a further generalization of Cartan Hartogs domains, one can define Hua constructions as cf. [16]) HC D := {w 1,...,w r, z) C m 1 C m r w j 2μ j D : < 1, μ N D z, z) ν j,ν j > 0}. j Consider mappings of the form w, z) = U 1 w 1 )φ 1 z),...,u r w r )φ r z), ϕz)), 4 1) where ϕz) AutD), U j w j ) Um j ) φ j z) = N Dz 0, z 0 ) ν j /2μ j N D z, z 0 ) ν j /μ j, with z 0 = ϕ 1 0). As in Lemma 3 1, one can show that mappings 4 1) form a subgroup of AutHC D ). A natural question is then whether AutHC D ) consists entirely of such mappings. While we do not know the answer to this question, we show below that a similar argument as in the previous section does not work for Hua constructions since Lemma 3 2 in general fails in this case. For simplicity clarity, we consider the case m = r = 2 write N for N D z, z). Assume that ν := ν 1 >ν 2 set δ = ν 1 ν 2. Then on the boundary of HC D,wehave w 1 2μ 1 + w 2 2μ 2 N δ N ν = )

11 On automorphism groups of generalized Hua domains 471 Let Z 0 = w 1,w 2, z 1, 0,, 0) 1 HC D s 1, s 2, t 1, 0,, 0) T Z0 HC D. Then we have μ 1 w 1 w 1 2μ1 1) s 1 + μ 2 w 2 w 2 2μ2 1) N δ s 2 + AN z1 t 1 = 0, 4 3) where A = w 2 2μ 2 δnδ 1 ν N ν 1. We want to show that the following quantity is possibly negative: L := μ 2 1 w 1 2μ 1 1) s μ 2 2 w 2 2μ 2 1) N δ s AN z1 z 1 + B N z1 2 ) t 1 2, 4 4) where B = w 2 2μ 2 δδ 1)N δ 2 νν 1)N ν 2. From 4 3), we have s 1 2 = μ2 2 w 2 4μ2 2 N 2δ s A 2 N z1 2 t C μ 2 1 w, 4 5) 1 4μ 1 2 where C = μ 2 w 2 2μ2 1) N δ A w 2 s 2 t 1 N z1 + w 2 s 2 t 1 N z1 ). Plugging 4 5) into 4 4) using 4 2), we get w 1 2μ 1 L = μ 2 2 w 2 4μ2 2 N 2δ s A 2 N z1 2 t C w 2 2μ 2 N δ N ν )μ 2 2 w 2 2μ2 1) N δ s AN z1 z 1 + B N z1 2 ) t 1 2 ). 4 6) Consider D = IV n for example. We have at Z 0 N = 1 z 1 2 ) 2, N z1 = 2 z 1 1 z 1 2 ), N z1 z 1 = z ) Plugging 4 7) into 4 6),we get w 1 2μ 1 L = μ 2 2 w 2 2μ2 1) N ν+δ s C + 4 z 1 2 w 2 2μ 2 ν δ) 2 N ν+δ 1 + 2A w 2 2μ 2 N δ N ν )) t ) Note that A = δ w 2 2μ 2 N δ 1 N ν 1 ) ν δ)n ν 1. We can then rewrite 4 8) as w 1 2μ 1 L = w 2 2μ 2 1) N ν+δ 1 E + 2N δ 1 w 2 2μ 2 N δ N ν )F, 4 9) where E = μ 2 1 z 1 2 )s 2 + 2ν δ) z 1 w 2 t 1 2 F = w 2 2μ 2 δ ν N ν δ 1 ) t 1 2 δμ 2 w 2 2μ 2 1) 1 z 1 2 ) w 2 z 1 s 2 t 1 + w 2 z 1 s 2 t 1 ). Choosing s 2 t 1 non-zero such that E = 0, we get F = δ w 2 2μ ν δ) z 1 2 ) ν1 z 1 2 ) ν δ 1 ) t 1 2. Since 0 < 1 z 1 2 < 1 δ 0, for ν δ big, we obviously have F > 0. Note that in 4 9) w 2 2μ 2 N δ N ν = w 1 2μ 1 < 0, thus we finally get L < 0. REFERENCES [1] H. AHN, J. BYUN J.-D. PARK. Automorphisms of the Hartogs type domains over classical symmetric domains. Internat. J. Math ), pages). [2] E. BEDFORD J. DADOK. Bounded domains with prescribed group of automorphisms. Comment. Math. Helv ), [3] E. BEDFORD S. PINCHUK. Domains in C 2 with noncompact groups of holomorphic automorphisms translated from Russian), Math. USSR-Sb ), [4] J. FARAUT, S.KANEYUKI, A.KORÁNYI Q. LU, G.ROOS. Analysis geometry on complex homogeneous domains. Progr. Math. 185 Birkhäuser, 1999),

12 472 FENG RONG [5] R. GREENE, K.-T.KIM S. KRANTZ. The geometry of complex domains. Progr. Math. 219 Birkhäuser, 2011). [6] R. GREENE S. KRANTZ. Invariants of Bergman geometry results concerning the automorphism groups of domains in C n. Proceedings of the 1989 Conference in Cetraro D. Struppa, ed.) 1991). [7] S. HELGASON. Differential Geometry, Lie Groups Symmetric Spaces Academic Press, Inc., 1978). [8] L. HUA. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Transl. Math. Monog. 6 Amer. Math. Soc., Providence, RI, 1979). [9] S. KOBAYASHI. Transformation Groups in Differential Geometry Springer-Verlag, New York- Heidelberg, 1972). [10] S. KRANTZ. Function Theory of Several Complex Variables. Amer. Math. Soc., Providence, RI, 2000). [11] R. NARASIMHAN. Several Complex Variables. Chicago Lectures in Math. Chicago University Press, Chicago-London, 1971). [12] J.-P. ROSAY. Sur une caractérisation de la boule parmi les domaines de C n par son groupe d automorphismes. Ann. Inst. Fourier Grenoble) ), [13] R. SAERENS W. ZAME. The isometry groups of manifolds the automorphism groups of domains. Trans. Amer. Math. Soc ), [14] A. WANG, W. YIN, L. ZHANG G. ROOS. TheKähler-Einstein metric for some Hartogs domains over symmetric domains. Sci. China Ser. A: Math ), [15] B. WONG. Characterization of the unit ball in C n by its automorphism group. Invent. Math ), [16] W. YIN. Lu Qi Keng conjecture Hua domain. Sci. China Ser. A: Math ),

ENGG1410-F Tutorial 6

ENGG1410-F Tutorial 6 Jianwen Zhao Department of Computer Science and Engineering The Chinese University of Hong Kong 1/16 Problem 1. Matrix Diagonalization Diagonalize the following matrix: A = [ ] 1 2 4 3 2/16 Solution The