Theorem 1. Let D be an irreducible bounded symmetric domain in C', n>= 2. Let F be a (nonuniform) lattice in Aut (D), ie a discrete subgroup of Aut (D) for which N:= D/F (is noncompact and) has. finite volume (wrt the locally symmetric metric induced from D). Suppose that a group F isomorphic to F (as an abstract group) acts as a discrete automorphism group on a contractible K (ihler manifold tVI. Assume that lQ/f has a finite singularity free cover M (ie M is a manifold) which is quasiprojective ie admits a compactification as a projective variety IVI and that~ I\M is of codimension at least three in~ l. Then] Q is biholomorphically equivalent to D, and f is conjugate to F in Aut (D).

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We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs.

This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and Cheng in [CY75].

Rigidity question have attracted much interest in the past. In the compact case, we have the famous work of Calabi and Vesentini [3] and Mostow [17]. Whereas Calabi and Vesentini proved a local version, namely that compact quotients of bounded symmetric domains admit no nontrivial deformations in case the domain is irreducible and of complex dimension at least 2, Mostow proved a global rigidity result, at the expense, however, of working only within the class of quotients of symmetric domains. Mostow's work is based on quasiconformal mappings. A different analytic approach was recently undertaken by Siu [22]. If M is a compact K~ ihler manifold diffeomorphic (or, more generally, homotopically equivalent) to a quotient N of an irreducible bounded symmetric domain, he studied a harmonic homotopy equivalence the existence of which is assured by the theorem of EeUs and Sampson, and demonstrated that