18 research outputs found
Asymptotically conical Calabi-Yau manifolds, I
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 needed in earlier work to ,
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 is .Comment: 27 pages, various corrections, final versio
Warped quasi-asymptotically conical Calabi-Yau metrics
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
Let be a toric K\"ahler-Einstein Fano manifold. We show that any toric
shrinking gradient K\"ahler-Ricci soliton on certain toric blowups of
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