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  1. ; ; ; ( , Geometric and Functional Analysis)
    Free, publicly-accessible full text available April 1, 2025
  2. ; ; ( , ArXivorg)
    Let D be a toric Kähler-Einstein Fano manifold. We show that any toric shrinking gradient Kähler-Ricci soliton on certain proper modifications 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. 
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    Accepted Manuscript Full Text Available
  3. ; ; ; ( , ArXivorg)
    We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of C×P1 at one point. This completes the classification of such solitons in two complex dimensions. 
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    Accepted Manuscript Full Text Available
  4. ; ; ( , ArXivorg)
    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. 
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    Accepted Manuscript Full Text Available