Uniqueness theorems of K�hler metrics of semipositive bisectional curvature on compact Hermitian symmetric spaces

Abstract

In [8] we studied metric rigidity theorems on Hermitian locally symmetric manifolds of negative Ricci curvature. In particular we proved that on a compact, locally symmetric and locally irreducible Hermitian manifold of negative Ricci curvature and of rank > 2, there exists only one Hermitian metric of seminegative curvature, namely the K~hler-Einstein metric, up to multiplicative constants. In the same vein one can ask if a similar theorem remains valid for compact Hermitian symmetric spaces of rank__> 2. In the present article we prove. Theorem 1. Let (X, g) be an irreducible compact Hermitian symmetric manifold of rank > 2. Suppose h is a twice differentiable Ktihler metric on X such that (X, h) carries semipositive holomorphic bisectional curvature. Then (X,h) is itself a Hermitian symmetric manifold. More precisely, there exists a biholomorphism 9 of X and a positive constant c such that h = cO*g.

The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature

“…In order to explain our approach, it is helpful to discuss the geometry of compact Hermitian symmetric manifolds and particularly the role played by certain special rational curves (cf. Mok [16]). Let (M, g) be an irreducible compact Hermitian symmetric manifold and let a: (M, g) <-+ (P N , Fubini-Study) be the first canonical (isometric) protective embedding of M. Let 2 be the collection of projective lines in P N that are already contained in M (which we identify with o(M)); they are totally geodesic in (M, g).…”

“…under consideration attains its minimum at e = 0. Expanding(15) we obtain(16) 2Re/^-= 0.Clearly one can replace e^ by e^e^ for any real 0 without changing the preceding argument. As a consequence, we obtainR tftf = ° forl ^ P ^ m >and thus m(17) *tftf = 0 for|e^) = I C V Now equation (17) implies by (6)(18) A > R m .Recall by (3) that A = (d/dt 2 )R(/l(t), W); x(0, xT^T), /8(0) = a, X (0) = f, and V t(/) j8(/) s vdt(Ox(O s «• Thus, for T, = 7?…”

Rigidity of proper holomorphic mappings between equidimensional bounded symmetric domains

“…Of particular importance is the phenomenon of rigidity of complex structures of such manifolds (e.g., see Siu [15,16]). In 1987, Mok [10,11] obtained metric rigidity theorems on Hermitian locally symmetric manifolds, and their proofs are applied to the study of holomorphic mappings between Hermitian locally symmetric manifolds of the same type, yielding various rigidity theorems on holomorphic mappings (see Mok [12]). Inspired by the idea in Mok's work [10,11], Tsai [17,18] We shall now present an outline of the argument in our proof of Theorem 1.1.…”

Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

“…In this work we generalize the idea of [4] and [5]. Their argument cannot be generalized to non-symmetric homogeneous spaces as Mok [16] has shown that other homogeneous spaces that are not covers of products of Hermitian Symmetric Spaces don't posses Kähler metrics with non-negative curvature. But complex G c /P , being a quotient of a compact Lie group, does posses a metric induced by the standard bi-invariant metric of the compact Lie group.…”