Uniqueness of some Calabi–Yau metrics on $${\mathbf {C}}^{{n}}$$

Higher regularity for singular Kähler-Einstein metrics

“…The main new technical ingredient in the proofs of these theorems is an estimate for solutions of the complex Monge-Ampère equation on non-collapsed balls in polarized Kähler manifolds with Ricci curvature bounds, or their Gromov-Hausdorff limits. This extends a related estimate from [34], where we considered balls that are Gromov-Hausdorff close to a metric cone of the form C× C(Y ), with smooth Y . Roughly speaking the result says that if a solution u of a Monge-Ampère equation with sufficiently small L ∞ norm concentrates near the (almost) singular set of such a ball, then the solution must decay by a definite amount when passing to a smaller ball.…”

“…In this result we do not assume, as we did in [34], that Y is smooth. In addition, note that on the right hand side of (2.2) the supremum of |u| is taken on the set B(p, 1) \ N δ which is typically larger than the set {r h > δ} ∩ B(p, 1) if Y has singularities.…”

“…We prove this result by proving successively more general cases. We start with the following, which follows the approach of [34,Proposition 4.5].…”

On partial uniqueness of complete non-compact Ricci flat metrics

“…See [13, 8.5 Theorem A4] for example. Geometric uniqueness of Ricci flat metrics, as in [5,6,9,10,11,13,16], usually involves both Monge-Ampere uniqueness and i∂∂−lemma under decay conditions. For recent work on Liouville theorem of Monge-Ampere equations on product manifolds, see Hein [12] for example.…”

Canonical Identification at Infinity for Ricci-Flat Manifolds

“…Ricci-flat manifolds with Euclidean volume growth are important objects in numerous areas of mathematics and physics, including Kähler and Sasaki geometry, general relativity, and string theory; see for instance [8][9][10][11][12][13][14][15][16][17][18] among others.…”