Metric rigidity theorems on Hermitian locally symmetric spaces.
Let X = Omega/Gamma be a compact quotient of an irreducible bounded symmetric domain Omega of rank >/=2 by a discrete group omega of automorphisms without fixed points. It is well known that the Kähler-Einstein metric g on X carries seminegative curvature (in the sense of Griffiths). I show that any Hermitian metric h on X carrying seminegative curvature must be a constant multiple of g. This can be applied to prove rigidity theorems of holomorphic maps from X into Hermitian manifolds (Y, k) carrying seminegative curvature. These results are also generalized to the case of quotients of finite volume. On the other hand, let (X(c), g(c)) be an irreducible compact Hermitian symmetric manifold of rank >/=2. Then g(c) is Kähler and carries semipositive holomorphic bisectional curvature. I prove that any Kähler h on X(c) carrying semipositive holomorphic bisectional curvature must be equal to g(c) up to a constant multiple and up to a biholomorphic transformation of X(c).
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Department of Mathematics, Columbia University, New York, NY 10027.