Expanding Ricci solitons coming out of weakly PIC1 metric cones

TL;DR

This paper proves that certain weakly PIC1 Ricci flows originating from metric cones are expanding gradient Ricci solitons, leading to a canonical biholomorphic equivalence of the cone at infinity with complex Euclidean space.

Contribution

It establishes that weakly PIC1 Ricci flows with Euclidean volume growth emerging from metric cones are necessarily expanding gradient Ricci solitons, and characterizes the asymptotic geometry of related Kähler manifolds.

Findings

Weakly PIC1 Ricci flows with Euclidean volume growth are expanding gradient Ricci solitons.

Metric cones at infinity of certain Kähler manifolds are biholomorphic to complex Euclidean space.

The results extend understanding of the structure of Ricci flows and their asymptotic geometry.

Abstract

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 K\"ahler manifold with Euclidean volume growth is biholomorphic to complex Euclidean space in a canonical way.