Complete cscK Metrics on the Local Models of the Conifold Transition

On complete constant scalar curvature Kähler metrics with Poincaré–Mok–Yau asymptotic property

“…In this case, we have seen that F (φ) > 0 when φ ∈ (a, +∞). From (7), we also see that when φ → +∞, the upper bound of t = t(φ) exists since the degree of F is n + 1. For simplicity, we take this upper bound to be zero since the solution ( 7) is unique up to be a constant.…”

“…In the following we will discuss more details of the solutions (7) for different signs of c and finish the proofs of Theorems 1.1 and 1.2.…”

“…Since the only root of F (φ) = 0 is a, F (φ) obtains its minimum at the point a and F (φ) > 0 for all φ > a. As φ → ∞, φ n−1 F (φ) → 1 φ and solution (7) has the property that t log φ. Hence φ = φ(t) is defined on the entire punctured space C n * .…”

“…Definition 1.1 (see also [7]). Let X be a compact Kähler manifold with a Kähler metric ω X and let S be a higher co-dimensional subvariety.…”

Nonexistence for complete Kähler–Einstein metrics on some noncompact manifolds

“…In the Ricci-flat case, this was answered in [22] based on a theorem proved in [19] for volume of complete noncompact Riemannian manifolds. In order to handle this question, the authors (in [5,4]) introduced the concept of complete metrics with Poincaré-Mok-Yau (PMY) asymptotic property, and constructed many constant scalar curvature Kähler metrics with PMY asymptotic property on some special types of noncompact Kähler manifolds. Since these PMY type metrics are not Kähler-Einstein, naturally one can ask the following question Problem 1.2.…”