Search for: All records

Let 𝐷 be a toric Kähler–Einstein Fano manifold. We show that any toric shrinking gradient Kähler–Ricci soliton on certain toric blowups of C×D satisfies a complex Monge–Ampère 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. 

We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci 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 Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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 Calabi-Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler 4-manifolds without twistor theory. 

In 1996, H.-D. Cao constructed a U(n)-invariant steady gradient Kähler-Ricci soliton on Cn and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on Cn is necessarily U(n)-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for n=2. Here, we construct a family of U(1)×U(n−1)-invariant, but not U(n)-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real (1,1)-forms (in particular, with strictly positive sectional curvature) on Cn for n≥3, thereby answering Cao's question in the negative for n≥3. This family of steady Ricci solitons interpolates between Cao's U(n)-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's U(n−1)-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of Pn endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by 2 on real (1,1)-forms. 

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 Hölder spaces. 

We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone.