18 research outputs found

    Asymptotically conical Calabi-Yau manifolds, I

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    This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition O(r−n−ϵ)O(r^{-n-\epsilon}) needed in earlier work to O(r−ϵ)O(r^{-\epsilon}), relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on T∗SnT^*S^n is −2nn−1-2\frac{n}{n-1}.Comment: 27 pages, various corrections, final versio

    Warped quasi-asymptotically conical Calabi-Yau metrics

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    We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. 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 blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted H\"older spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact K\"ahler form.Comment: 64 pages, added the construction of singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, thus interpolating between two different Calabi-Yau cone

    An Aubin continuity path for shrinking gradient K\"ahler-Ricci solitons

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    Let DD be a toric K\"ahler-Einstein Fano manifold. We show that any toric shrinking gradient K\"ahler-Ricci soliton on certain toric blowups of C×D\mathbb{C}\times D satisfies a complex Monge-Amp\`ere 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.Comment: 66 pages, various corrections, Proposition 7.15 revise
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