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**Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains.**
*(English)*
Zbl 0616.53040

This article is one of a series of articles by the author on the metric rigidity of Hermitian locally symmetric spaces of rank \(\geq 2\). These results have been announced in [Proc. Natl. Acad. Sci. USA 83, 2288-2290 (1986; Zbl 0593.53034)]. These metric rigidity results have analogues in real differential geometry for locally symmetric manifolds of rank \(\geq 2\) and nonpositive sectional curvature.

Define a Hermitian manifold (X,h) to have negative (seminegative) curvature if the holomorphic tangent bundle T(X) is negative (seminegative) in the sense of Griffiths with respect to the induced Hermitian metric. For Kaehler manifolds (X,h) this is equivalent to the negativity (seminegativity) of the holomorphic bisectional curvature. However, in general the Hermitian connection differs from the Riemannian connection if (X,h) is not Kaehler. The main result of this article is the following Theorem 1. Let (X,g) be a locally symmetric Hermitian manifold of finite volume uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Let h be a Hermitian metric on X such that (X,h) carries seminegative curvature and h is dominated by a constant multiple of g. Then \(h=cg\) for some constant \(c>0.\)

Using the strong rigidity theorem of Siu the author then obtains Theorem 2. Let (M,h) be a compact Kaehler manifold of seminegative holomorphic bisectional curvature that is homotopic to a compact Hermitian locally symmetric manifold (X,g) uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Then (M,h) is up to a constant factor on h biholomorphically or conjugate-biholomorphically isometric to (X,g).

Theorem 1 also yields as a corollary a rigidity theorem for holomorphic mappings from X into a seminegatively curved Hermitian manifold. Theorem 3. Let (X,g) be a compact locally symmetric Hermitian manifold uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Let (N,h) be a Hermitian manifold carrying seminegative curvature. If \(f: X\to N\) is a nonconstant holomorphic mapping, then up to a multiplicative constant f is an isometry.

As corollaries of these main results the author also derives the following: a) There exists no nontrivial homomorphism from the fundamental group of a compact irreducible Hermitian locally symmetric manifold of rank \(\geq 2\) and negative Ricci curvature into the fundamental group of a strongly negatively curved compact Kaehler manifold. b) If \(\Omega\) is a bounded symmetric domain admitting a complete Hermitian metric of bounded torsion and of negative curvature bounded between 2 negative constants, then \(\Omega\) is of rank 1.

Define a Hermitian manifold (X,h) to have negative (seminegative) curvature if the holomorphic tangent bundle T(X) is negative (seminegative) in the sense of Griffiths with respect to the induced Hermitian metric. For Kaehler manifolds (X,h) this is equivalent to the negativity (seminegativity) of the holomorphic bisectional curvature. However, in general the Hermitian connection differs from the Riemannian connection if (X,h) is not Kaehler. The main result of this article is the following Theorem 1. Let (X,g) be a locally symmetric Hermitian manifold of finite volume uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Let h be a Hermitian metric on X such that (X,h) carries seminegative curvature and h is dominated by a constant multiple of g. Then \(h=cg\) for some constant \(c>0.\)

Using the strong rigidity theorem of Siu the author then obtains Theorem 2. Let (M,h) be a compact Kaehler manifold of seminegative holomorphic bisectional curvature that is homotopic to a compact Hermitian locally symmetric manifold (X,g) uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Then (M,h) is up to a constant factor on h biholomorphically or conjugate-biholomorphically isometric to (X,g).

Theorem 1 also yields as a corollary a rigidity theorem for holomorphic mappings from X into a seminegatively curved Hermitian manifold. Theorem 3. Let (X,g) be a compact locally symmetric Hermitian manifold uniformized by an irreducible bounded symmetric domain of rank \(\geq 2\). Let (N,h) be a Hermitian manifold carrying seminegative curvature. If \(f: X\to N\) is a nonconstant holomorphic mapping, then up to a multiplicative constant f is an isometry.

As corollaries of these main results the author also derives the following: a) There exists no nontrivial homomorphism from the fundamental group of a compact irreducible Hermitian locally symmetric manifold of rank \(\geq 2\) and negative Ricci curvature into the fundamental group of a strongly negatively curved compact Kaehler manifold. b) If \(\Omega\) is a bounded symmetric domain admitting a complete Hermitian metric of bounded torsion and of negative curvature bounded between 2 negative constants, then \(\Omega\) is of rank 1.

Reviewer: P.Eberlein

### MSC:

53C35 | Differential geometry of symmetric spaces |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |