arXiv subjects
Publications and source records attributed to Luke T. Peachey.
We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p) \times O(q)$ symmetry in dimension $p+q$ for any pair of integers $p,q \geq 2$ and discuss their asymptotic geometry.
The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper we use the recent uniqueness theory in this context, also developed by the second author and H. Yin, to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: every complete Ricci flow on a surface over a time interval $(0,\varepsilon)$ admits a $t\downarrow 0$ limit within the class of admissible initial data. This makes surfaces the first nontrivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals $(0,T)$, and a class of initial data that induces them.
Motivated by recent work of Deruelle-Schulze-Simon, we study complete weakly PIC1 Ricci flows with Euclidean volume growth coming out of metric cones. We show that such a Ricci flow must be an expanding gradient Ricci soliton, and as a consequence, any metric cone at infinity of a complete weakly PIC1 Kähler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.