Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay.

*(English)*Zbl 0358.32006
Ann. Math. (2) 105, 225-264 (1977); erratum ibid. 109, 621-623 (1979).

The uniformization theorem of Blemann surfaces can be understood in differential geometry in the following manner:

(i). Every compact surface with positive curvature is conformally equivalent to the Riemann sphere.

(ii). Every noncompact complete surface with positive curvature is conformally equivalent to the complex line.

(iii). Every simply connected (complete) surface with curvature bounded from above by a negative constant is conformally equivalent to the open unit 1-disc.

A natural higher-dimensional analog of these three facts would be that in all cases we replace surfaces by complete Kähler manifolds and

in Case (i) the Biemann sphere by a complex projective space,

in Case (ii) the complex line by a complex Euclidean space, and

in Case (iii) the open unit 1-disc by a bounded domain in a complex Euclidean space.

Using the \(L^2\) estimates of \(\bar{\partial}\) and some results from differential geometry such as comparison theorems, sub-mean-value properties, and volume estimates, the authors prove the following main theorem: If \(M\) is a simply connected, complete, Kähler manifold of complex dimension \(n\) with

\[ 0\geq \mathrm{sectional curvature} \geq -\frac{A}{r^2+\epsilon} \]

where \(A\) and \(\epsilon\) are positive numbers and \(r\) is the distance from a fixed point of \(M\), then \(M\) is biholomorphic to \(\epsilon\).

(i). Every compact surface with positive curvature is conformally equivalent to the Riemann sphere.

(ii). Every noncompact complete surface with positive curvature is conformally equivalent to the complex line.

(iii). Every simply connected (complete) surface with curvature bounded from above by a negative constant is conformally equivalent to the open unit 1-disc.

A natural higher-dimensional analog of these three facts would be that in all cases we replace surfaces by complete Kähler manifolds and

in Case (i) the Biemann sphere by a complex projective space,

in Case (ii) the complex line by a complex Euclidean space, and

in Case (iii) the open unit 1-disc by a bounded domain in a complex Euclidean space.

Using the \(L^2\) estimates of \(\bar{\partial}\) and some results from differential geometry such as comparison theorems, sub-mean-value properties, and volume estimates, the authors prove the following main theorem: If \(M\) is a simply connected, complete, Kähler manifold of complex dimension \(n\) with

\[ 0\geq \mathrm{sectional curvature} \geq -\frac{A}{r^2+\epsilon} \]

where \(A\) and \(\epsilon\) are positive numbers and \(r\) is the distance from a fixed point of \(M\), then \(M\) is biholomorphic to \(\epsilon\).

Reviewer: D. Ghişa