• Corpus ID: 15037503

Numerical Kaehler-Ricci soliton on the second del Pezzo

@article{Headrick2007NumericalKS,
  title={Numerical Kaehler-Ricci soliton on the second del Pezzo},
  author={Matthew Headrick and Toby Wiseman},
  journal={arXiv: Differential Geometry},
  year={2007}
}
The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density). 
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References

SHOWING 1-10 OF 15 REFERENCES

Geometry of Ricci Solitons*

Ricci solitons are natural generalizations of Einstein metrics on one hand, and are special solutions of the Ricci flow of Hamilton on the other hand. In this paper we survey some of the recent…

Numerical Kähler-Einstein Metric on the Third del Pezzo

The third del Pezzo surface admits a unique Kähler-Einstein metric, which is not known in closed form. The manifold’s toric structure reduces the Einstein equation to a single Monge-Ampère equation…

Kahler-Ricci solitons on toric manifolds with positive first Chern class

Gaussian densities and stability for some Ricci solitons

In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central…

A new holomorphic invariant and uniqueness of Kähler-Ricci solitons

Abstract. In this paper, a new holomorphic invariant is defined on a compact Kähler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the…

Convergence of Kähler-Ricci flow

In this paper, we prove a theorem on convergence of Kahler-Ricci flow on a compact Kahler manifold which admits a Kahler-Ricci soliton

The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group

On utilise la methode de continuite pour demontrer l'existence de metriques de Kahler-Einstein pour des varietes de Kahler de fibre linge positif anticanonique sous l'hypothese additionnelle de…

Kähler-Einstein metrics on complex surfaces withC1>0

AbstractVarious estimates of the lower bound of the holomorphic invariant α(M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed…

The entropy formula for the Ricci flow and its geometric applications

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…

On Rotationally Symmetric Hamilton’s Equation for Kähler–Einstein Metrics