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Singularities in U (2)-invariant 4d Ricci flow

  • Author(s): Appleton, Alexander James
  • Advisor(s): Bamler, Richard
  • Wilkening, Jon
  • et al.
Abstract

Firstly, we analyze the steady Ricci soliton equation for a certain class of metrics on complex line bundles over K\"ahler-Einstein manifolds of positive scalar curvature. We show that these spaces admit a non-collapsed steady gradient Ricci soliton metric. In the four (real-) dimensional case, this yields a new family of non-collapsed steady Ricci solitons on the complex line bundles $O(-k)$, $k \geq 3$, over $\mathbb{C}P^1$. These solitons are $U(2)$-invariant, non-K\"ahler, and asymptotic to the the quotient of the four dimensional Bryant soliton by $\Z_k$. As a byproduct of our work we also find Taub-Nut like Ricci solitons on $\R^4$ and demonstrate a new proof for the existence of the Bryant soliton.

Secondly, we investigate the formation of singularities in four dimensional $U(2)$-invariant Ricci flow and show that the Eguchi-Hanson space can occur as a blow-up limit. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time and the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\R^4 / \Z_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\Z_2$, and (iv) the shrinking cylinder $\R \times \R P^3$. It also follows from our work that in four dimensional Ricci flow an embedded two dimensional sphere of any self-intersection number $k \in \Z$ may collapse to a point in finite time and thereby produce a singularity. For $|k|\geq 3$ the singularities we construct are of Type II, yielding a new infinite family of Type II singularities. Numerical simulations indicate that their blow-up limits are the four dimensional steady Ricci solitons described above.

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