Automorphism groups of domains that depend on fewer than the maximal number of parameters
aa r X i v : . [ m a t h . C V ] O c t Automorphism Groups of Domainsthat Depend on Fewer Than theMaximal Number of Parameters by Jisoo Byun and Steven G. Krantz Let Ω ⊆ C n be a domain, that is a connected, open set. Let Aut(Ω) denotethe collection of biholomorphic self-mappings of Ω (see [ISK1] or [GRK] fora survey of this topic). This set forms a group when equipped with thebinary operation of composition of mappings. In case Ω is bounded, thenthe group is in fact a real (never a complex) Lie group. We call this groupthe automorphism group of Ω.It is naturally a matter of considerable interest to describe the automor-phism group for a given domain Ω. In the best of all possible circumstances,we would like to give an explicit description of this group. As an instance,in case Ω = B ≡ { ( z , · · · , z n ) ∈ C n : | z | + · · · + | z n | < } , then the automorphism group of Ω is generated by (i) the unitary rotationsand (ii) the M¨obius transformations y ( z , z , . . . , z n ) z − a − az , q − | a | z − az , . . . , q − | a | z n − az for a ∈ C , | a | <
1. It is worth noting (as this is part of the theme of thepresent paper) that any description of the automorphisms of B must involveall n variables. Such a statement is already true for the unitary group alone. The second author thanks the American Institute of Mathematics for its hospitalityand support during a portion of this work. Both authors thank the Banach Center inWarsaw, Poland for its hospitality during a recent conference during which this problemwas discussed. The second author was supported in part by a grant from the Dean ofthe Graduate School at Washington University and a grant from the National ScienceFoundation.
1t is not true that the automorphism group depends only on n − n − There are other important domains—ones that are currently a focus ofconsiderable study—whose automorphism groups are much simpler. For theKohn-Nirenberg domain (see [JIS], [KON]), the automorphisms consist of rotations in one variable only . The purpose of the present paper is to producea geometric criterion which will guarantee that the automorphism group of agiven domain Ω ⊆ C n will depend on fewer than the full number of variablesin the ambient space. This result will simplify the example in [JIS], and willalso provide further examples for the future. We indicate some of these latterexamples at the end of the present paper. We are happy to thank Peter Pflug for a helpful conversation.
Fix a pseudoconvex domain Ω ⊆ C with smooth boundary. Because of thediscussion in the first section, we may suppose that Ω is not (biholomorphi-cally) the ball. In case Ω is strongly pseudoconvex, we may then conclude bya theorem of Bun Wong and Rosay (see [KRA1]) that Aut(Ω) is compact (infact we shall make this a standing hypothesis in the discussion that follows).Let ( z, w ) be the coordinates in C . We take it that the origin = (0 , ∂ Ω. Now suppose that, near , Ω is defined by the inequality ρ ( z, w ) ≡ Re w + ϕ ( z, z ) + ψ ( z, z, Im w ) · (Im w ) < . ( ∗ )In what follows we shall always use the letter ρ to denote a defining functionfor Ω. Applying a unitary rotation if necessary, we may arrange that thecomplex tangent space H at is equal to { ( z,
0) : z ∈ C } .Let Aut (Ω) denote the automorphisms of Ω that fix . We shall assumebelow that H ∩ Ω is an open subset of H . If f is an automorphism of Clearly these statements can be formulated in terms of the dimension of the auto-morphism group. We leave this task as an exercise for the reader. For the purposes here,the formulation in terms of the ambient complex variables is more convenient and moreaccurate. In the paper [GIK], it is shown that if Ω is a hyperbolic domain and the dimension ofthe automorphism group exceeds n + 2 then the domain must be a ball. The result of thepresent paper is in a philosophically similar spirit, but the restrictions on the dimensionare more severe. e f ( ζ ) = f ( ζ ,
0) for ( ζ , ∈ H ∩ Ω and f = ( f , f ). Define F e f ( z, w ) = ( e f ( z ) , w ).Now we shall list the standing hypotheses that will be in place for theremainder of this paper: Standing Hypotheses1.
The set H ∩ Ω is an open subset of H . The function ϕ in the definition of ρ has no harmonic terms. If f ∈ Aut (Ω) then F e f ∈ Aut (Ω). The automorphism group Aut (Ω) is compact. The domain Ω is complete hyperbolic. Any automorphism of Ω continues analytically to a neighborhood of ∈ ∂ Ω.Standing Hypothesis perhaps merits some discussion. In case ϕ has harmonic terms, we may write ϕ ( z, z ) = ∞ X k =2 a k z k + ∞ X k =2 a k z k + e ϕ ( z, z ) , where e ϕ has no harmonic term. Let µ ( z ) = P k a k z k be holomorphic. Thenthe holomorphic coordinate change( e z, e w ) = ( z, w + 2 X k a k z k )defines a new local defining functionRe e w + e ϕ ( z, z ) + ψ ( z, z, Im w ) · (Im w ) < . Note that the lead term here has no harmonic term.Now our main result is this: 3 heorem 1
Let Ω be a domain as described above. Then any automorphism f ( z ) = ( f ( z ) , f ( z )) fixing the origin of Ω must have the form f ( z ) = ( ϕ ( z ) , z ) . In other words, any automorphism of Ω fixing the origin will depend only onthe first variable (and not on the second). If f ∈ Aut(Ω) then certainly ρ ◦ f is also a local defining function for Ωnear . Therefore ρ ◦ f ( z,
0) = µ · ρ ( z,
0) ( ∗∗ )for some positive function µ near . As a result, we may conclude (seeStanding Hypothesis ) that the quadratic part of µ · ρ ( z,
0) has no harmonicterms. This observation also applies to the lefthand side of ( ∗∗ ), so we seethatRe f ( z, ϕ ( f ( z, , f ( z, ψ ( f ( z, , f ( z, , Im f ( z, · (Im f ( z, has no harmonic terms. CLAIM:
We assert that f ( z, ≡ Proof of the Claim:
If not, then f ( z,
0) = X k ≥ a k z k . Let a k be the nonzero coefficient with least index.Since ϕ has no harmonic terms and f ( z,
0) = b z + X k ≥ b k z k with b = 0, we see that ϕ ( f , f ) = X k,ℓ b k b ℓ z k z ℓ . Certainly b = 0, hence ϕ ( f , f ) has no harmonic terms.Now ψ ( f ( z, , f ( z, , Im f ( z, · (Im f ( z, k as the first nonvanishing term. Hence the firstharmonic term Re a k z k ≡ . But this implies that a k ≡
0. And that is a contradiction.Now, as a consequence of the claim, we certainly know that f ( H ∩ Ω) ⊆ H . As a result, f | H ∩ Ω is an automorphism of H ∩ Ω that fixes . In our earliernotation, e f ∈ Aut( H ∩ Ω).Referring to Standing Hypothesis , we now consider F g f − ◦ f ∈ Aut (Ω) . We see that F g f − ◦ f ( z,
0) = F g f − ( e f ( z ) , z, . Writing Φ ≡ F g f − ◦ f , we may say that Φ( z,
0) = ( z, d Φ( z,
0) = a ( z )0 b ( z ) ! for some a ( z ) , b ( z ) that are holomorphic on H . Of course d Φ takes the realtangent space at to the real tangent space at and the complex tangentspace at to the complex tangent space at . Since Aut (Ω) is compact, weconclude that sup | b ( z ) | = 1 and b (0) = 1 hence (by the maximum principle) b ≡
1. Finally, since Ω is complete hyperbolic, a ( z ) ≡
0. We conclude that d Φ( z,
0) = ! , the identity matrix. By the Cartan Uniqueness Theorem, we conclude thatΦ( z ) = z for all z ∈ Ω.Therefore, we finally arrived that f ( z, w ) = ( e f ( z ) , w ) . In short, f depends on fewer than the maximal number of parameters.5 .1 Levi Flat Domains Now we present a variant of our main result.
Standing Hypotheses1.
The local defining function for Ω at the origin is Re w = 0. Any automorphism of Ω continues analytically to a neighborhood of ∈ ∂ Ω. The automorphism group Aut (Ω) is compact. The domain Ω is complete hyperbolic. If f ∈ Aut (Ω) then F e f ∈ Aut (Ω).Fix f ∈ Aut (Ω). By the same argument as in the Main Theorem, wehave that ρ ◦ f ( z,
0) = µρ ( z, , where ρ = Re w . This implies that Re f ≡ f = ( f , f ). Weobtain that f preseve that H ∩ Ω. By Standing Hypotheses 5, we considerΦ = F ˜ f − ◦ f . Then Φ( z,
0) = ( z, f : d Φ( z,
0) = ! . By the power series expansion of Φ,Φ ( z, w ) = z + w X a jk z j w k Φ ( z, w ) = w + w X b jk z j w k . By Standing Hypotheses 1, there is a positive δ such that ( z, it ) is in theboundary of Ω, for all | z | < δ and real number | t | < δ . So, we get Φ( z, it )lies in the boundary of Ω. This implies thatRe (cid:16) it + ( it ) X b jk z j ( it ) k (cid:17) = 0 .
6e consider the left hand side as a power series with respect to t . For all j, k ≥
0, Re ( − b jk z j i k ) = 0 . For each j, k , we can choose two different complex numbers α, β such that α j is pure imaginary and β j is real. Note that j and k are fixed. We canassume that i k is a real number ±
1. We get thatRe ( − b jk z j ) = 0 . The above equation holds for all sufficient small complex numbers. We caninsert α, β into it. This implies that b jk is a real and pure imaginary number.Hence, all b jk are zero. We arrive at the conclusion thatΦ ( z, w ) = z + w X a jk z j w k Φ ( z, w ) = w. We want to prove that Φ (0 , w ) = w P a k w k is indentically zero. Ex-pecting a contradiction, we assume that a k is the first nonzero term. Weconsider the N -times composition of Φ restricted to (0 , w ). By a calculation,Φ N (0 , w ) = ( N a k w k + (higherorderterms) , w ) . Since Φ N is a precompact family, we arrive at a contradiction.Therefore, Φ(0 , w ) = (0 , w ). Take a derivative at (0 , w ). Then d Φ(0 , w ) = a ( w ) b ( w )0 1 ! . By the same technique as in the last section, we have the identity matrix at(0 , w ). We can apply Cartan’s Uniqueness Theorem. Finally we get that Φis the identity map of Ω. This means that f ( z, w ) = ( ˜ f ( z ) , w ). EXAMPLE 1
First let us look at the Kohn-Nirenberg domain [KON],which is given (in the complex variables z, w ) byRe w + | zw | + | z | + 157 | z | Re z < .
7t is a simple matter to verify that this domain satisfies the hypothesis about S in our Theorem. Also the domain is of finite type, so every automorphismextends smoothly to the boundary. We may conclude immediately that anyautomorphism fixing the origin depends on just one variable. Then simplecalculations show (see [JIS]) that the only automorphisms are rotations inthe z variable. EXAMPLE 2
Let ϕ be a C ∞ function on R , even, nonnegative, supportedin the interval [ − / , / − / , / { ( z , z ) ∈ C : (1 − ϕ ( | z | ))(1 − ϕ ( | − z | )) · [ − | z | ]+ ϕ ( | z | ) ϕ ( | − z | ) · [ − ǫ + Re z ] < } . Then Ω is nothing other than the unit ball in C with a flat bump centeredabout the spherical boundary point (1 , − ǫ,
0) has a neighborhood in the boundary that lies in the hyperplane { Re z = 1 − ǫ } . It is straightforward to see that the only automorphisms ofΩ are rotations in the z variable (see [LER]).Now examine our main theorem. This domain Ω satisfies the hypothesesof that theorem with the origin replaced by (1 − ǫ, − ǫ,
0) forces standing hypothesis to hold. So thisexample is an illustration of our main result. The automorphism group onlydepends on the z variable. This is the first paper to explore the questions posed here. There is clearlya need for a result in all dimensions, and for results that have more flexiblehypotheses.It would certainly be of interest to have concrete examples to which ourresults do (or do not) apply. The theorem definitely does not apply to thecomplex ellipsoids E m,n = { ( z , z ) : | z | m + | z | n < } m, n ∈ N . And of course they should not. The result also does not applyto the ball or the Siegel upper half space. More, they do not apply to anyof the bounded symmetric domains of Cartan (see [GRK] for a discussion ofsome of these specialized domains).What would be ideal is to have a theorem that, given 0 < k < n ∈ N , characterizes domains in C n whose automorphisms depend only on k variables. This will be the subject of future investigations.9 eferences [BEP] E. Bedford and S. Pinchuk, Domains in C with non-compact holomor-phic automorphism group (translated from Russian), Math. USSR-Sb. [BER]
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