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A New Complete Two-Dimensional Shrinking Gradient Kähler-Ricci Soliton

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Abstract

We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of \(\mathbb{C}\times \mathbb{P}^{1}\) at one point. This completes the classification of such solitons in two complex dimensions.

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Notes

  1. Note that in this paper we are using the Ricci flow equation \(\partial _{t} g(t) = - \operatorname{Ric}_{g(t)}\) which is more common in the Kähler setting. By a simple reparameterization of the time parameter, any such flow can be converted to a Ricci flow satisfying \(\partial _{t} g(t) = - 2\operatorname{Ric}_{g(t)}\), which is the subject of [Bam23, Bam202].

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Acknowledgements

The authors wish to thank Song Sun and Jeff Viaclovsky for useful discussions.

Funding

The first author is supported by NSF grant DMS-1906500. The second author is supported by the grant Connect Talent “COCOSYM” of the région des Pays de la Loire and the Centre Henri Lebesgue, programme ANR-11-LABX-0020-0, the third author is supported by NSF grant DMS-1906466, and the fourth author is supported by grants ANR-17-CE40-0034 of the French National Research Agency ANR (Project CCEM) and ANR-AAPG2020 (Project PARAPLUI).

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Authors and Affiliations

  1. Department of Mathematics, University of California Berkeley, Berkeley, CA, 94720, USA

    Richard H. Bamler

  2. Mathematics Department, Stony Brook University, Stony Brook, NY, 11794, USA

    Charles Cifarelli

  3. Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, 75080, USA

    Ronan J. Conlon

  4. CNRS, Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, 91405, Orsay, France

    Alix Deruelle

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  1. Richard H. Bamler
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  2. Charles Cifarelli
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  4. Alix Deruelle
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Correspondence to Ronan J. Conlon.

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Bamler, R.H., Cifarelli, C., Conlon, R.J. et al. A New Complete Two-Dimensional Shrinking Gradient Kähler-Ricci Soliton. Geom. Funct. Anal. 34, 377–392 (2024). https://doi.org/10.1007/s00039-024-00668-9

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