Non-negatively curved Kähler manifolds with average quadratic curvature decay

Abstract

491-530.], we prove that the universal cover M of M is biholomorphic to C n provided either that (M, g) has average quadratic curvature decay, or M supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomorphic bisectional curvature. We also classify certain local limits arising from the Kähler-Ricci flow in the absence of uniform estimates on the injectivity radius.

On the simply connectedness of nonnegatively curved Kähler manifolds and applications

“…Then by the monotonicity of the eigenvalues of the Ricci curvature in t [13] (Theorem 6.1), the Ricci curvature at p is uniformly bounded below away from zero for all t. On the other hand, by the LYH type differential inequality by Cao [6], we know that d dt R(p, t) ≥ 0. Hence as in [12], for any t i → ∞, (M, g(t i ), p) will subconverge to a gradient Kähler-Ricci soliton with positive Ricci curvature and nonnegative holomorphic bisectional curvature. Hence as in the proof of Theorem 2.4, we can prove that there exist T > 0 and some a > 0 such that for t ≥ T there exists q(t) ∈ B t (p, a) which is the unique point satisfying ∇ t R(q(t), t) = 0 in B t (p, 3a).…”

“…By passing to a subsequence still denoted by t i , we can find 0). Thus Z i = f i is a holomorphic vector field, and it can be shown that the gradient of f is zero at the origin (see [12] for example), which is an isolated zero because f ij = R h ij > 0. In fact by its construction in [12, §2] and [6], there exists 0 < r 1 < r such that Z is the unique solution of the equation…”

“…Let R be the scalar curvature and let (1.1) k(r, x) = 1 V x (r) B x (r) R be the average of the scalar curvature over the geodesic ball with radius r and center at x. In [11], [12] and [13] the authors obtained, among other things, the following uniformization results:…”

“…g(t) exists for −∞ < t < ∞) or that tR is uniformly bounded in space-time. Note that either condition will imply that eigenvalues of Ric(p, t) are uniformly pinched; see [12] for example.…”

“…Remark 2.1. Theorem 2.2 was proved by the authors in [12] under the stronger assumption that tR(p, t) is uniformly bounded for all t ≥ 0 independently of p. In particular, it was proved there that M is biholomorphic to C n . We point out that the injectivity radius bound in Theorem 2.2 (i) allows a direct application of the methods in [11], and thus a more direct proof of this result.…”