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Automorphisms of certain Stein manifolds. (English) Zbl 0829.32009
Let $$(M,g)$$ be a compact, real analytic Riemannian manifold. Let $$0<r\leq\infty$$. Denote by $$T^rM$$ the disk bundle over $$M$$ that consists of all the tangent vectors having $$g$$-length smaller than $$r$$. For a given $$(M,g)$$ there exists an $$r$$ and a canonically defined (adapted to the metric $$g$$) complex manifold structure $$J_g$$ on $$T^rM$$. The $$g$$-norm square function $$\rho$$ is strictly plurisubharmonic with respect to $$J_g$$, hence $$(T^rM,J_g)$$ is a Stein manifold. Denote by $$\kappa_g$$ the Kähler metric on $$T^rM$$ induced by $$\rho$$. The main purpose of this paper is to study biholomorphic maps between and automorphisms of this type of Stein manifolds. We also discuss the question when such a biholomorphism is an isometry with respect to $$\kappa_g$$.

##### MSC:
 32E10 Stein spaces 32W20 Complex Monge-Ampère operators 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
##### Keywords:
biholomorphic maps; automorphisms; Stein manifolds; isometry
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