Metric rigidity theorems on Hermitian locally symmetric spaces
AUTOR(ES)
Mok, Ngaiming
RESUMO
Let X = Ω/Γ be a compact quotient of an irreducible bounded symmetric domain Ω of rank ≥2 by a discrete group ω 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 (Xc, gc) be an irreducible compact Hermitian symmetric manifold of rank ≥2. Then gc is Kähler and carries semipositive holomorphic bisectional curvature. I prove that any Kähler h on Xc carrying semipositive holomorphic bisectional curvature must be equal to gc up to a constant multiple and up to a biholomorphic transformation of Xc.
