8月18日-报告8-卢金-Proper mappings between indefinite hyperbolic spaces and type I classical domains.pdf

Introduction and background Indefinite hyperbolic spaces Type I classical domains Proper mappings between indefinite hyperbolic spaces and type I classical domains Jin Lu School of Internet, Anhui University Work with X. Huang X. Tang and M. Xiao August, 2022 Shanghai University Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 1 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Contents • Introduction and background • Indefinite hyperbolic spaces • Type I classical domains Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 2 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Given integers n ≥ 2 and 0 ≤ l ≤ n − 1, the generalized complex unit ball is defined as the following domain in Pn : Bnl = {[z0 , ..., zn ] ∈ Pn : |z0 |2 + ... + |zl |2 > |zl+1 |2 + ... + |zn |2 }. For 0 ≤ k ≤ m, let Ik ,m be the m × m diagonal matrix, where its first k diagonal elements equal −1 and the rest equal 1. Denote by SU(l + 1, n + 1) the special indefinite unitary group that consists of matrices A ∈ SL(n + 1, C) satisfying t AIl+1,n+1 A = Il+1,n+1 . Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 3 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains The generalized ball Bnl possesses a canonical indefinite metric ωBnl that is invariant under the action of its automorphism group SU(l + 1, n + 1): l n X X √ ωBnl = − −1∂ ∂¯ log( |zj |2 − |zj |2 ). j=0 j=l+1 The generalized ball equipped with the above indefinite metric is often called an indefinite hyperbolic space form. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 4 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Theorem (A,Baouendi-Ebenfelt-Huang, 2011) Let N ≥ n, 1 ≤ l ≤ n−1 , 1 ≤ l 0 ≤ N−1 and 1 ≤ l ≤ l 0 < 2l. Let U be an open subset in 2 2 Pn containing some p ∈ ∂Bnl with U ∩ Bnl being connected, and F a holomorphic map from U into PN . Assume F (U ∩ Bnl ) ⊆ BNl0 and F (U ∩ ∂Bnl ) ⊆ ∂BNl0 . Then F is an isometric embedding from (U ∩ Bnl , ωBnl ) into (BNl0 , ωBN0 ). l Here we say F is isometric if it preserves the indefinite hyperbolic metrics: F ∗ (ωBN0 ) = ωBnl on U ∩ Bnl . l M. S. Baouendi, P. Ebenfelt, X. Huang, Holomorphic mappings between hyperquadrics with small signature difference, Amer. J. Math. 133 (6) (2011) 1633-1661. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 5 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains By using a different approach that utilizes structure of the moduli space of linear subspaces contained in generalized balls, Ng establishes the global version of Theorem. Theorem (B,Ng, 2013) Let 1 ≤ l < n2 , 1 ≤ l 0 < N2 and f : Bnl → BNl0 be a proper holomorphic map. If l 0 ≤ 2l − 1, then f extends to a linear embedding of Pn into PN . S. Ng, Proper holomorphic mappings on flag domains of SU(p, q)−type on projective spaces, Michigan Math. J., 62 (2013) 769-777. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 6 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Main result Theorem (1) Let N ≥ n ≥ 3 , 1 ≤ l ≤ n − 2, l ≤ l 0 ≤ N − 1. Let U be an open subset in Pn containing some p ∈ ∂Bnl and F be a holomorphic map from U into PN . Assume U ∩ Bnl is connected and F (U ∩ Bnl ) ⊆ BNl0 , F (U ∩ ∂Bnl ) ⊆ ∂BNl0 . Assume one of the following conditions holds: (1). l 0 < 2l, l 0 < n − 1; (2). l 0 < 2l, N − l 0 < n; (3). N − l 0 < 2n − 2l − 1, l 0 < n − 1; (4). N − l 0 < 2n − 2l − 1, N − l 0 < n. Then F is an isometric embedding from (U ∩ Bnl , ωBnl ) to (BNl0 , ωBN0 ). l Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 7 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Definition (1) Let F be a holomorphic rational map from Pn to PN . Write I ⊆ Pn for the set of indeterminacy of F . We say F is a rational proper map from Bnl to BNl0 , if F maps from Bnl \ I to BNl0 and maps ∂Bnl \ I to ∂BNl0 . Theorem (1) can be immediately applied to study rational proper maps between generalized balls. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 8 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Corollary Let N ≥ n ≥ 3 , 1 ≤ l ≤ n − 2, l ≤ l 0 ≤ N − 1. Assume one of the conditions in (1)–(4) of Theorem (1) holds. Let F be a rational proper map from Bnl to BNl0 . Then F is a linear embedding from Pn to PN . Moreover, there exists h ∈ Aut(BNl0 ) such that h ◦ F ([z]) = [z0 , ..., zl , 0, ..., 0, zl+1 , ..., zn , 0, ..., 0], for [z] = [z0 , ..., zl , zl+1 , ..., zn ] ∈ Pn , where the first zero tuple has l 0 − l components. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 9 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Note if l ≥ 1, then every proper holomorphic map from Bnl to BNl0 extends to a rational map from Pn to PN (see [Ng1]). Thus Corollary still holds if we assume F is a proper holomorphic from Bnl to BNl0 instead of assuming it is a rational proper map from Bnl to BNl0 . Hence Corollary has Theorem (B) as its special case. It also has Corollary 1.6 in [BEH] as its special case. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 10 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Theorem (1) is optimal in the sense that it fails if none of the conditions (1)–(4) holds. Indeed, suppose all of the conditions (1)–(4) fail. Then one of the following two cases must hold: • (A). l 0 ≥ 2l and N − l 0 ≥ 2n − 2l − 1; • (B). N − l 0 ≥ n and l 0 ≥ n − 1. The next two examples show the conclusion in Theorem (1) fails in each of the cases. Example (1) corresponds to the case (A) with l 0 = 2l and N − l 0 = 2n − 2l − 1. Example (2) corresponds to the case (B) with N − l 0 = n and l 0 = n − 1. Furthermore, the map in Example (1) is indeed a rational proper map between the generalized balls in the sense of Definition. Thus it also shows Corollary fails if none of the conditions (1)–(4) holds. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 11 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Example (1) (Generalized Whitney map from Bl+k to B2l+2k−1 ) Let l ≥ 1, k ≥ 1. Write l 2l [w, z] = [w0 , w1 , · · · , wl , z1 , · · · , zk ] for the homogeneous coordinates of Pl+k and l k X X  Bll+k = [w, z] ∈ Pk +l : |wi |2 > |zj |2 . i=0 j=1 Write U = Pk +l \ {w0 = zk = 0}. Consider the following map G : U → P2k +2l−1 : G([w, z]) = [w02 , w0 w1 , · · · , w0 wl , w1 zk , w2 zk , · · · , wl zk , w0 z1 , w0 z2 , · · · , w0 zk−1 , z1 zk , z2 zk , · · · , zk−1 zk , zk2 ]. Write the above components on the right hand side as G1 , · · · , G2k +2l and set P2l+1 P 2 |G|22l+1 = − i=1 |Gi |2 + 2k+2l j=2l+2 |Gj | . Notice that P P |G|22l+1 = (|w0 |2 + |zk |2 )(− li=0 |wi |2 + kj=1 |zj |2 ). Consequently, G maps U ∩ Bl+k to l −1 B2l+2k−1 and maps U ∩ ∂Bll+k to ∂B2l+2k . Hence the statement in Theorem (1) fails in 2l 2l this case. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 12 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Example (2) (Generalized Whitney map from Bl+k to B2l+2k−1 l l+k −1 ) Let l ≥ 1, k ≥ 1. Let the homogeneous coordinates [w, z] and Bl+k ⊆ Pl+k be the same as in Example 6. Let l V = Pl+k \ {w0 = wl = 0} and H : V → P2k +2l−1 be defined as follows: H([w, z]) = [w02 , w0 w1 , · · · w0 wl−1 , wl z1 , wl z2 , · · · , wl zk , w0 z1 , w0 z2 , · · · , w0 zk , w1 wl , w2 wl , · · · , wl2 ]. Write the above components on the right hand side as H1 , · · · , H2k +2l and set P P2k +2l 2 2 |H|2l+k = − l+k i=1 |Hi | + j=l+k+1 |Hj | . Notice that P P |H|2l+k = (|w0 |2 − |wl |2 )(− li=0 |wi |2 + kj=1 |zj |2 ). Thus H maps V ∩ ∂Bl+k to l l+k ∂B2l+2k−1 to l+k−1 . In particular, set V+ := {[w, z] ∈ V : |w0 | > |wl |}. Then H maps V+ ∩ Bl B2l+2k−1 and maps V+ ∩ ∂Bl+k to ∂B2l+2k−1 l l+k−1 l+k−1 . Hence the statement in Theorem fails in 2l+2k −1 this case. This map H is, however, not a rational proper map from Bl+k to Bl+k in l −1 the sense of Definition, as it maps some point in Bl+k to P2l+2k−1 \ B2l+2k−1 l l+k−1 . Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 13 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Lemma Let l, m, a, b be nonnegative integers such that m ≥ 2, 1 ≤ l ≤ m − 1. Let ϕ1 , ..., ϕa , ψ1 , ..., ψb be homogeneous holomorphic polynomials of the same degree in Cm such that − a X j=1 2 |ϕj (z)| + b X |ψj (z)|2 = A(z, z̄)|z|2l , z ∈ Cm , (1) j=1 where A(z, z̄) is a real polynomial. Assume one of the following conditions holds: (1). a < l, a < m − l; (2). a < l, b < l; (3). b < m − l, a < m − l; (4). b < m − l, b < l. Then A(z, z̄) ≡ 0. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 14 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Recall for 0 ≤ l ≤ n − 1, the generalized Siegel upper-half space is defined by Snl = {(z, w) ∈ Cn−1 × C : Im(w) > n−1 X δj,l |zj |2 }. j=1 Its boundary is the standard hyperquadrics: P 2 0 Hnl = {(z, w) ∈ Cn−1 × C : Im(w) = n−1 j=1 δj,l |zj | }. Similarly for l ≤ l ≤ N − 1, we define SNl,l 0 ,n = {(Z , W ) ∈ CN−1 × C : Im(W ) > N−1 X δj,l,l 0 ,n |Zj |2 }. j=1 And SNl0 , HNl0 , HNl,l 0 ,n are all defined in a similar manner. Now for (z, w) = (z1 , · · ·, zn−1 , w) ∈ Cn , let Ψ(z, w) = [i + w, 2z, i − w] ∈ Pn . Then Ψ is the Cayley transformation which biholomorphically maps Snl and its boundary Hnl onto Bnl \ {[z0 , · · ·, zn ] : z0 + zn = 0} and ∂Bnl \ {[z0 , · · ·, zn ] : z0 + zn = 0}, respectively. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 15 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Composing F with automorphisms of Bnl and BNl0 if necessary, we assume that q0 = [1, 0, ..., 0, 1] ∈ ∂Bnl and F (q0 ) = [1, 0, ..., 0, 1] ∈ ∂BNl0 . Denote by Ψ the aforementioned Cayley transformation from Snl to Bnl , and Φ the Cayley transformation e := Φ−1 ◦ F ◦ Ψ is well-defined in a small neighborhood of from SN 0 to BN0 . Then F l,l ,n l 0 ∈ Hnl ; and F̃ is side-preserving (i.e., it maps Snl to SNl,l 0 ,n near 0). Moreover, by the definition of the geometric rank (see Section 3 in [HLTX]), to show F is of geometric e has zero geometric rank near 0. rank zero near q0 , it suffices to prove the new map F X. Huang, J. Lu, X. Tang, M. Xiao, Boundary characterization of holomorphic isometric embeddings between indefinite hyperbolic spaces, Adv. Math., 374 (2020) 107388. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 16 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains e . That is, F is To keep notations simple, we will still write the new map as F instead of F now a holomorphic map from a neighborhood V of 0 ∈ Hnl to CN , satisfying F (V ∩ Snl ) ⊆ SNl,l 0 ,n and F (V ∩ Hnl ) ⊆ HNl,l 0 ,n . By shrinking V if necessary, we can additionally assume M1 := V ∩ Hnl is connected and F is CR transversal along M1 . Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 17 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Next for each p ∈ M1 , we associate it with a map Fp defined as: Fp = τpF ◦ F ◦ σp0 . (2) Here σp0 ∈ Aut(Hnl ) and τpF ∈ Aut(HNl,l 0 ,n ) are as defined in [HLTX]. Then Fp is a holomorphic map in a neighborhood of 0 ∈ Cn , which sends an open piece of Hnl into HNl,l 0 ,n with Fp (0) = 0. Moreover, Fp (U ∩ Snl ) ⊆ SNl,l 0 ,n . Let Fp∗ , Fp∗∗ be the first and second normalizations of Fp . Then, Fp∗∗ map 0 to 0, and maps Hnl (respectively, Snl ) to ∗∗ ∗∗ HNl,l 0 ,n (respectively, SNl,l 0 ,n ) near 0. Write Fp∗∗ = (fp∗∗ , φ∗∗ p , gp ), where fp has n − 1 ∗∗ components, φ∗∗ p has N − n components, and gp is a scalar function. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 18 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Lemma. (Lemma 2.2 in [BH]) For each p ∈ M1 , Fp∗∗ satisfies the normalization condition:  ∗∗(1)  fp∗∗ = z + 2i ap (z)w + Owt (4)   ∗∗(2) φ∗∗ (z) + Owt (3) p = φp    g ∗∗ = w + O (5), p wt with ∗∗(1) hz̄, ap ∗∗(2) (z)il |z|2l = |φp (z)|2τ , τ = l 0 − l. M. S. Baouendi, X. Huang, Super-rigidity for holomorphic mappings between hyperqadrics with positive signature, J. Differential Geom. 69 (2) (2005) 379-398. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 19 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains ∗∗(1) We briefly recall the notion of geometric rank. If we write ap (z) = zA(p) for any (n − 1) × (n − 1) matrix A(p), then the geometric rank of F at p is defined as the rank of the matrix A(p). In particular, F have geometric rank zero at p if and only if A(p) is the zero matrix. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 20 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains ∂ 2 (f ∗∗ ) The rank of the (n − 1) × (n − 1) matric A(p) = −2i( ∂z p∂w k |0 )1 0}. An important step toward understanding proper maps between type I classical domains was due to Ng [Ng2] where he first found its deep connection with mapping problems for proper maps between generalized balls. In such a way, he would be able to apply results in CR geometry to mapping problems between bounded symmetric domains. Among other things, he proved that every proper holomorphic map f : DrI ,s → DrI 0 ,s is standard (i.e., totally geodesically isometric embedding, up to normalization constants, with respect to the Bergman metrics) if s ≥ r ≥ 2 and r 0 ≤ min{2r − 1, s}. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 25 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains In a recent nice paper of Chan [Ch], he posed the following conjecture, that was inspired by the work of Kim-Zaitsev [KZ2], Kim [K] and his own investigation in [Ch]. The statement in the conjecture would generalize the aforementioned theorem of Ng when r 0 < s. S. T. Chan, Rigidity of proper holomorphic maps between type-I irreducible bounded symmetric domains, Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373. S. Ng, Holomorphic Double Fibration and the mapping problems of Classical Domains, International Mathematics Research Notices, 2 (2015) 291-324. S. Kim, D. Zaitsev, Rigidity of CR maps between Shilov boundaries of bounded symmetric domains, Invent. Math., 193 (2) (2013) 409-437. S. Kim, Proper holomorphic maps between bounded symmetric domains, In complex analysis and geometry, (2015) 207-219. S. Kim, D. Zaitsev, Rigidity of proper holomorphic maps between bounded symmetric domains, Math. Ann., 362 (1-2) (2015) 639-677. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 26 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Conjecture I Let f : Dp,q → DpI 0 ,q 0 , p ≥ q > 1 be a proper holomorphic map. Assume q 0 < p and one of the following conditions holds: (1) p0 < 2p − 1; (2) q 0 < 2q − 1. Then (I). p0 ≥ p, q 0 ≥ q. I (II). Moreover, after composing with suitable automorphisms of Dp,q and DpI 0 ,q 0 , f takes the following form: f :z→ z 0 0 h(z) ! . (3) I Here h is a certain holomorphic (p0 − p) × (q 0 − q)−matrix valued function on Dp,q t I satisfying that Iq 0 −q − h h is positive definite on Dp,q . Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 27 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains I In the remaining context of the paper, as in [Ch], if a proper map f : Dp,q → DpI 0 ,q 0 satisfies the conclusion of Conjecture (i.e., it takes the form (3) after composing with automorphisms), then we say f is of diagonal type. It is known that Conjecture holds under the additional assumption that f extends smoothly to a neighborhood of a smooth boundary point. This is a consequence of results obtained by Kim-Zaitsev [KZ2] and Kim [K]. (See Corollary 1 in [KZ2] for case (1), and Theorem 1.2 in [K] for case (2)). Moreover, the assumption in (1) or (2) of Conjecture cannot be weakened. Indeed, Seo (see page 445 in [S1]) constructed a proper holomorphic map I (generalized Whitney map) from DrI ,s to D2r −1,2s−1 , which is not of diagonal type. A. Seo, New examples of proper holomorphic maps among symmetric domains, Michigan Math. J. 64 (2) (2015) 435-448. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 28 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains By further developing the double fibration ideas introduced in [Ng2], Chan himself [Ch] confirmed part (I) of Conjecture. He also proved part (II) of Conjecture under the condition in (2), while still left open part (II) under the condition in (1). See Theorem 1.3 in [Ch]. We give a complete affirmative answer to Conjecture under the condition in (1). Thus our result together with the work of Chan [Ch] leads to the following theorem: Theorem (2) Conjecture holds. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 29 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains I Let f : Dq,p → DqI 0 ,p0 be a holomorphic map. Say f is fibral-image-preserving with respect to the double fibrations: 1 πq,p 2 πq,p I I Dq,p ←− Pq−1 × Dq,p −→ Dq,p , πq10 ,p0 0 πq20 ,p0 Dq 0 ,p0 ←− Pq −1 × DqI 0 ,p0 −→ DqI 0 ,p0 . Here 1 I πq,p ([X ], Z ) = [X , XZ ]q , for [X ] ∈ Pq−1 , Z ∈ Dq,p . 2 And πq,p is the standard projection onto Dq,p . Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 30 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains I For x = [A, B]q ∈ Dq,p ⊆ Pq+p−1 and Z ∈ Dq,p , their fibral images are defined, respectively, as the following:
2 1 I x ] = [A, B]]q = πq,p (πq,p )−1 ([A, B]q ) ⊆ Dq,p ;
1 2 Z ] = πq,p (πq,p )−1 (Z ) ⊆ Dq,p . As shown in [Ng2], we indeed have the following formulas for the fibral images: I x ] = {Z ∈ Dq,p : AZ = B}; Z ] = {[A, AZ ]q ∈ Dq,p : [A] ∈ Pq−1 }. If for any [A, B]q ∈ Dq,p , we have f ([A, B]]q ) ⊆ [C, D]]q 0 for some [C, D]q 0 ∈ Dq 0 ,p0 . Furthermore, let U be an open subset of Dq,p . Say a holomorphic map g : U → Dq 0 ,p0 is a moduli map of f on U if f ([A, B]]q ) ⊆ g([A, B]q )] for all [A, B]q ∈ U. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 31 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Proposition (1) I Let f : Dq,p → DqI 0 ,p0 be a proper holomorphic map where p ≥ q ≥ 2, and 3 ≤ q 0 < p. Then the following statements hold. (a). Then f is fibral-image-preserving with respect to the double fibrations. And there exists a holomorphic map g : U ⊆ Dq,p → Dq 0 ,p0 such that g is a moduli map of f on U, where U is a dense open subset of Dq,p . Furthermore, g extends to a rational map 0 0 from Pp+q−1 to Pp +q −1 . Write I for the set of indeterminacy of g, we have g(∂Dq,p \ I) ⊆ ∂Dq 0 ,p0 . And we have p0 ≥ p, q 0 ≥ q. (b). We have g maps Dq,p \ I to Dq 0 ,p0 . Consequently, g is a rational proper map from Dq,p to Dq 0 ,p0 . (c). If either p0 < 2p − 1 or q 0 < 2q − 1, then f is of diagonal type. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 32 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains We prove the following consequence of [Ng2, Ch]. The result generalizes a theorem of Tu [T2]. Note when r = s − 1, Proposition 13 is reduced to Theorem 1.1 in [T2]. It also generalizes Theorem 1.3 of [Ng2] in the case r 0 = s. Proposition (2) I Let s > r ≥ 2. Then every proper holomorphic map f : DrI ,s → Ds,s is standard. That is, f is a totally geodesic isometric embedding (up to normalization constants) with respect to the Bergman metrics. Z. Tu, Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains, Mathematische Zeitschrift, (2002) 13-35. S. Ng, Holomorphic Double Fibration and the mapping problems of Classical Domains, International Mathematics Research Notices, 2 (2015) 291-324. Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 33 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains Corollary I There exist no proper holomorphic mappings from DrI ,s+1 to Ds,s for s ≥ r ≥ 2. Note Corollary fails if r = 1 and s ≥ 2. Indeed there is a proper holomorphic map from I I D1,s+1 = Bs+1 to Ds,s (see [T2]). Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 34 / 35 Introduction and background Indefinite hyperbolic spaces Type I classical domains The End Thank you! Jin Lu (Anhui University) Proper mappings between indefinite hyperbolic spaces and type I classical domains 35 / 35