Remarks on holomorphic isometric embeddings between bounded symmetric domains
aa r X i v : . [ m a t h . C V ] M a r REMARKS ON HOLOMORPHIC ISOMETRIC EMBEDDINGSBETWEEN BOUNDED SYMMETRIC DOMAINS
SHAN TAI CHAN
Abstract.
In this article, we study holomorphic isometric embeddings betweenbounded symmetric domains. In particular, we show the total geodesy of anyholomorphic isometric embedding between reducible bounded symmetric domainswith the same rank. Introduction
In [Mok12], Mok established an (algebraic) extension theorem of germs of holo-morphic isometric embeddings between bounded symmetric domains with respect totheir Bergman metrics up to a normalizing constant. It is known from the proof ofthe Hermitian Metric Rigidity Theorem on Hermitian locally symmetric spaces (cf.[Mok89]) that such a holomorphic isometric embedding f is totally geodesic when-ever all irreducible factors of the domain of f are irreducible bounded symmetricdomains of rank ≥ n -ball, n ≥
2, to a product of complex unit ballswith respect to the canonical K¨ahler metrics. Recently, the author, Xiao and Yuan[CXY17] have generalized this result, namely, we have proven the total geodesy ofany holomorphic isometric embedding from a product of complex unit n j -balls for1 ≤ j ≤ k , and n j ≥ j , into a product of complex unit balls with respect tothe canonical K¨ahler metrics.In general, there are nonstandard (i.e., not totally geodesic) holomorphic isometricembeddings from a complex unit ball into a bounded symmetric domain of rank ≥ f , we can still obtain some rigidity results for f . In particular, we consider holomorphic isometric embeddings between boundedsymmetric domains with the same rank. In the present article, we denote by g D thecanonical K¨ahler-Einstein metric on an irreducible bounded symmetric domain D ⋐ C n normalized so that the minimal disks of D are of constant Gaussian curvature −
2. We refer the readers to Mok-Tsai [MT92] for the notion of minimal disks ofbounded symmetric domains. The main result of the present article is the following.
Theorem 1.1 (Main Theorem) . Let D = D × · · · × D k and Ω = Ω × · · · × Ω m be bounded symmetric domains in their Harish-Chandra realizations such that rank(Ω) = rank( D ) ≥ , where D j , ≤ j ≤ k , Ω l , ≤ l ≤ m , are irreducible Mathematics Subject Classification.
Primary 53C55, 32H02, 32M15. bounded symmetric domains. Let F : D → Ω be a holomorphic isometric embeddingfrom ( D , λ g D ) × · · · × ( D k , λ k g D k ) to (Ω , µ g Ω ) × · · · × (Ω m , µ m g Ω m ) for somepositive real constants λ j , ≤ j ≤ k , and µ l , ≤ l ≤ m . Then, F is totally geodesic.Remark. If we further make the assumption that all D j , 1 ≤ j ≤ k , are of rank ≥ Preliminary
In the present article, we will always assume that a given bounded symmetricdomain Ω ⋐ C N is in its Harish-Chandra realization. Let Ω = Ω × · · · × Ω m ⋐ C N be a bounded symmetric domain, where Ω j ⋐ C n j , 1 ≤ j ≤ m , are irreduciblebounded symmetric domains. Then, it well-known that the Bergman kernel of Ω isgiven by K Ω ( Z, ξ ) = m Y j =1 K Ω j ( Z j , ξ j ) , where K Ω j ( Z j , ξ j ) is the Bergman kernel of Ω j for 1 ≤ j ≤ m , Z = ( Z , . . . , Z m ) and ξ = ( ξ , . . . , ξ m ) for Z j , ξ j ∈ C n j , 1 ≤ j ≤ m . In addition, for 1 ≤ j ≤ m , we alsohave K Ω j ( Z j , ξ j ) = C j h Ω j ( Z j , ξ j ) − ( p (Ω j )+2) for some positive real constant C j and for some polynomial h Ω j ( Z j , ξ j ) in ( Z j , ξ j ),where p (Ω j ) is some positive integer depending only on Ω j (cf. [CM17]). Givenan irreducible bounded symmetric domain D ⋐ C n , we let g D be the canonicalK¨ahler-Einstein metric on D normalized so that minimal disks in D are of constantGaussian curvature − ω g D the corresponding K¨ahler form of ( D, g D ).Then, from [CM17] we have ω g D = −√− ∂∂ log h D ( w, w ) , where w ∈ D ⊂ C n are the Harish-Chandra coordinates. Moreover, we denote by ds U the Bergman metric of any bounded domain U ⋐ C n . We have the followinglemma, which is one of the consequences of [CXY17, Theorem 4.25]. Lemma 2.1. (cf. Theorem 4.25 in [CXY17] ) Let F = ( F , . . . , F m ) : ( D , λ g D ) ×· · · × ( D k , λ k g D k ) → (Ω , λ ′ g Ω ) × · · · × (Ω m , λ ′ m g Ω m ) be a holomorphic isometry, i.e., m X j =1 λ ′ j F ∗ j g Ω j = k M l =1 λ l g D l , where D l , ≤ l ≤ k , Ω j , ≤ j ≤ m , are irreducible bounded symmetric domains intheir Harish-Chandra realizations, and λ l , ≤ l ≤ k , λ ′ j , ≤ j ≤ m , are positivereal constants. Then, F : D × · · · × D k → Ω × · · · × Ω m is a proper holomorphicmap. Holomorphic isometric embeddings between bounded symmetricdomains with the same rank
Motivated by the Polydisk Theorem (cf. [Mok89, Wo72]), we first study holomor-phic isometric embeddings from (cid:0) ∆ r , ds r (cid:1) to (Ω , g Ω ) for any irreducible bounded OLOMORPHIC ISOMETRIC EMBEDDINGS 3 symmetric domain of rank r ≥
2. Then, we obtain the following result, which is aspecial case of our main result, i.e., Theorem 1.1.
Theorem 3.1.
Let F : (cid:0) ∆ r , ds r (cid:1) → (Ω , g Ω ) be a holomorphic isometric embed-ding, where Ω ⋐ C N is an irreducible bounded symmetric domain of rank r ≥ .Then, F is totally geodesic.Proof. We prove by induction on the rank r of the target irreducible bounded sym-metric domain Ω.Firstly, we consider the case where rank(Ω) = 2. By Mok-Tsai [MT92, Proposi-tion 2.2], the radial limit F ∗ ( e iθ , w ) exists almost everywhere on ∂ ∆ × ∆. Writing F θ ( w ) := F ∗ ( e iθ , w ), Mok-Tsai [MT92, pp. 103-104] showed that F θ is holomorphicin w by the Cauchy integral formula. Then, F θ (∆) lies in a maximal face of Ω, whichis biholomorphic to a complex unit ball B n (Ω) , where n (Ω) is the null dimension ofΩ (see [Mok89, p. 105]). As in Mok-Tsai [MT92] and Tsai [Ts93, Proposition 1.1],one further deduces from the Fatou’s Theorem that F ( z, w ) = 12 π √− Z ∂ ∆ F ∗ ( ξ, w ) ξ − z dξ (cf. Mok [Mok07]) and thus for each z ∈ ∆, F z (∆) lies in some maximal charac-teristic symmetric subspace Ω ′ ∼ = B n (Ω) of Ω, where F z ( w ) := F ( z , w ). Moreover,(Ω ′ , g Ω | Ω ′ ) ∼ = ( B n (Ω) , g B n (Ω) ) is of constant holomorphic sectional curvature −
2. Then,for each z ∈ ∆, we have a holomorphic isometry F z : (∆ , g ∆ ) → (Ω ′ , g Ω | Ω ′ ) ∼ =( B n (Ω) , g B n (Ω) ), which is totally geodesic by the Gauss equation since both the do-main and the target are of constant holomorphic sectional curvature −
2. Denoteby σ the (1 , F (∆ ) , g Ω | F (∆ ) ) in (Ω , g Ω ).Write ∆ := ∆ × ∆ with ∆ j = ∆, j = 1 ,
2. This shows that for any tangentvector α ∈ T w (∆ ) ⊂ T ( z,w ) (∆ ), we have σ ( α ′ , α ′ ) = 0, where α ′ := dF ( z,w ) ( α ).Similarly, for any tangent vector β ∈ T z (∆ ) ⊂ T ( z,w ) (∆ ), we have σ ( β ′ , β ′ ) = 0,where β ′ := dF ( z,w ) ( β ). Moreover, we have σ ( α ′ , β ′ ) = 0. These together imply thatthe second fundamental form σ vanishes identically and thus F is totally geodesic.The theorem is proved when the rank of the target irreducible bounded symmetricdomain is equal to r = 2.Assume that the statement of the theorem holds true when the target irreduciblebounded symmetric domain is of rank r − ≥ r ≥
3. Then, we considerthe case where Ω is of rank r ≥
3. The same arguments imply that for each z ∈ ∆,the map F z : ∆ r − → Ω defined by F z ( z , . . . , z r ) := F ( z , z , . . . , z r ) has the imagelying inside a maximal characteristic symmetric subspace Ω ′ of Ω. It is well-knownfrom [Wo72] that Ω ′ is an irreducible bounded symmetric domain of rank ( r − F z : (∆ r − , ds r − ) → (Ω ′ , g Ω ′ ) for z ∈ ∆.By the induction hypothesis, F z is totally geodesic for each z ∈ ∆. Applying thesimilar arguments as before, we see that F is totally geodesic. (cid:3) Now, we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Write F := ( F , . . . , F m ), where F l : D → Ω l , 1 ≤ l ≤ m , areholomorphic maps. From the assumption, we have P ml =1 µ l F ∗ l g Ω l = L kj =1 λ j g D j . Itfollows from Lemma 2.1 that F is a proper holomorphic map. For simplicity, wewrite (Ω , g ′ Ω ) = (Ω , µ g Ω ) × · · · × (Ω m , µ m g Ω m ). Denote by σ the (1 , F ( D ) , g ′ Ω | F ( D ) ) in (Ω , g ′ Ω ). OLOMORPHIC ISOMETRIC EMBEDDINGS 4
If rank( D ) = rank(Ω) = 1, then D ∼ = B n and Ω ∼ = B m for some positive integers n and m . Note that for any λ > N ≥
1, ( B N , λg B N ) is of constantholomorphic sectional curvature − λ . Considering the holomorphic isometry F :( B n , λ g B n ) → ( B m , µ g B m ), we have − λ ≤ − µ and thus 0 < λ µ ≤ F that λ µ isa positive integer (see [CM17, Lemma 3]). Therefore, we have λ µ = 1, i.e., λ = µ .For any unit vector α ∈ T , x ( F ( B n )) and any x ∈ F ( B n ), we have k σ ( α, α ) k = R αααα ( B m , µ g B m ) − R αααα ( F ( B n ) , µ g B m | F ( B n ) ) = − µ − (cid:18) − λ (cid:19) = 0because λ = µ . Therefore, we have σ ( α, α ) = 0. Then, for any α ′ , β ′ ∈ T , x ( F ( B n )), x ∈ F ( B n ), we have0 = σ ( α ′ + β ′ , α ′ + β ′ ) = σ ( α ′ , α ′ ) + 2 σ ( α ′ , β ′ ) + σ ( β ′ , β ′ ) = 2 σ ( α ′ , β ′ )so that σ ( α ′ , β ′ ) = 0, i.e., σ ≡
0. Hence, F is totally geodesic.On the other hand, if rank( D j ) ≥ ≤ j ≤ k , then F is totally geodesicby Mok [Mok12, Theorem 1.3.2]. From now on, we assume that D is reducible,rank( D ) = rank(Ω) ≥ D is of rank 1, equivalentlybiholomorphic to some complex unit ball. That means we may write D = B q × · · · × B q k × D k +1 ×· · ·× D k , where D j , k +1 ≤ j ≤ k , are irreducible bounded symmetricdomains of rank at least 2 for some integer k ≥
1. For k + 1 ≤ j ≤ k , there is atotally geodesic complex submanifold M j of D j such that M j ∼ = ∆ r j − × B q j , where r j := rank( D j ) and q j denotes the complex dimension of the rank-1 characteristicsymmetric subspace of D j for k + 1 ≤ j ≤ k (cf. [MT92, Ts93]). Inductively,the arguments in the proof of Theorem 3.1 show that the (proper) holomorphicisometry F induces a holomorphic isometry f := F | B q ×···× B qk × B qk ×···× B qk from( B q , λ g B n ) × · · · × ( B q k , λ k g B qk ) to (Ω ( k ) , g ′ Ω | Ω ( k ) ) for some rank- k characteristicsymmetric subspace Ω ( k ) of Ω.Let q ′ l be the complex dimension of the rank-1 characteristic symmetric subspaceof Ω l for 1 ≤ l ≤ m . By the same arguments as in the proof of Theorem 3.1, themap f further induces holomorphic isometries from ( B q j , λ j g B qj ) to ( B q ′ lj , µ l j g B q ′ lj )for some l j and for any j , 1 ≤ j ≤ k . Similarly, f induces holomorphic isometriesfrom ( B q j , λ j g B qj ) to ( B q ′ lj , µ l j g B q ′ lj ) for some l j and for any j , k + 1 ≤ j ≤ k . Inaddition, we have λ j = µ l j , 1 ≤ j ≤ k , by the functional equation and the Gaussequation as before. Then, these induced holomorphic isometries from ( B q j , λ j g B qj )to ( B q ′ lj , µ l j g B q ′ lj ) are totally geodesic by the arguments as before. More precisely,for 1 ≤ j ≤ k , let ι j : B q j ֒ → B q × · · · × B q k be the standard embedding ι j ( Z j ) =( Z , . . . , Z j − , Z j , Z j +10 , . . . , Z k ) for fixed Z l ∈ B q l , 1 ≤ l ≤ k and l = j . Then, f ◦ ι j is totally geodesic for 1 ≤ j ≤ k and thus f is totally geodesic.For each x = F ( Z , . . . , Z k , W k +1 , . . . , W k ) ∈ S := F ( D ), we have σ ( α ′ j , α ′ j ) = 0 ∀ α ′ j = dF ( Z,W ) ( α j ) , α j ∈ T , Z j ( B q j ) ⊂ T , Z,W ) ( D ) , for 1 ≤ j ≤ k , where ( Z, W ) = ( Z , . . . , Z k , W k +1 , . . . , W k ) ∈ D = B q × · · · × B q k × D k +1 × · · · × D k . On the other hand, from [Mok12] we have σ ( β ′ j , β ′ j ) = 0 ∀ β ′ j = dF ( Z,W ) ( β j ) , β j ∈ T , W j ( D j ) ⊂ T , Z,W ) ( D ) , OLOMORPHIC ISOMETRIC EMBEDDINGS 5 because rank( D j ) ≥ k + 1 ≤ j ≤ k . Moreover, from the arguments in the proofof [Mok12, Theorem 1.3.2] we have σ ( η ′ i , η ′ j ) = 0for i = j , 1 ≤ i, j ≤ k , where η ′ µ ∈ T , F ( Z,W ) ( F ( D )) is the image of some tangent vectorof the µ -th direct factor D µ of D for 1 ≤ µ ≤ k . Hence, the second fundamentalform vanishes identically and thus F is totally geodesic. (cid:3) Through the discussion with Professor Wing-Keung To, we actually raised thefollowing problem about the structure of holomorphic isometric embeddings from areducible bounded symmetric domain to an irreducible bounded symmetric domainof higher rank.
Problem 3.2.
Let D = D × · · · × D k be a reducible bounded symmetric domain and Ω ⋐ C N be an irreducible bounded symmetric domain of rank ≥ , where D j , ≤ j ≤ k , are the irreducible factors of D . Let F : D → Ω be a holomorphic isometricembedding from ( D , g D ) × · · · × ( D k , g D k ) to (Ω , g Ω ) . Then, does F ( D ) lie insidesome totally geodesic product subdomain (Π , g Ω | Π ) ∼ = (Ω ′ , g ′ Ω ′ ) := (Ω ′ , g Ω ′ ) × · · · × (Ω ′ k , g Ω ′ k ) in Ω such that rank(Ω ′ ) = rank(Ω) ? More precisely, can F be factorizedas F ( Z , . . . , Z k ) = ι ( ˆ F ( Z ) , . . . , ˆ F k ( Z k )) for ( Z , . . . , Z k ) ∈ D × · · · × D k , where ˆ F j : ( D j , g D j ) → (Ω ′ j , g Ω ′ j ) , ≤ j ≤ k , areholomorphic isometries and ι : (Ω ′ , g ′ Ω ′ ) ֒ → (Ω , g Ω ) is the totally geodesic holomorphicisometric embedding?Remark. Problem 3.2 has been solved by Mok [Mok12] if all irreducible factors D j of D are of rank ≥ F is totally geodesic in thiscase. However, some irreducible factor D j of D could be of rank 1 in Problem 3.2,i.e., D j ∼ = B n j for some positive integer n j . In addition, Problem 3.2 is solved whenΩ is of rank 2 by Theorem 1.1. Acknowledgements
The main part of this work was done during the period that the author visitedThe University of Hong Kong in the summer of 2017. The author would like toexpress his gratitude to Professors Ngaiming Mok and Wing-Keung To for helpfuldiscussions on the topic of holomorphic isometric embeddings between (reducible)bounded symmetric domains.
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