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Free, publiclyaccessible full text available April 1, 2025

We construct many new examples of complete CalabiYau metrics of maximal volume growth on certain smoothings of Cartesian products of CalabiYau cones with smooth crosssections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blowup for manifolds with corners. A key analytical step in the construction of these CalabiYau metrics is to derive good mapping properties of the Laplacian on some suitable weighted Hölder spaces.more » « less

Let D be a toric KählerEinstein Fano manifold. We show that any toric shrinking gradient KählerRicci soliton on certain proper modifications of C×D satisfies a complex MongeAmpère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method.more » « less

We show that the underlying complex manifold of a complete noncompact twodimensional shrinking gradient KählerRicci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the FeldmanIlmanenKnopf conjecture for finite time Type I singularities of the KählerRicci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.more » « less

A Riemannian cone (C,gC) is by definition a warped product C=R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a CalabiYau cone if gC is a Ricciflat Kähler metric and if C admits a gCparallel holomorphic volume form; this is equivalent to the crosssection (L,gL) being a SasakiEinstein manifold. In this paper, we give a complete classification of all smooth complete CalabiYau manifolds asymptotic to some given CalabiYau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyperKähler 4manifolds without twistor theory.more » « less

We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient KählerRicci soliton in every Kähler class of an equivariant crepant resolution of a CalabiYau cone converging at a polynomial rate to Cao's steady gradient KählerRicci soliton on the cone.more » « less