An asymptotically cusped three dimensional expanding gradient Ricci soliton

TL;DR

This paper constructs a unique three-dimensional expanding gradient Ricci soliton on R x T^2 with asymptotic cusped and flat ends, under negative curvature pinching conditions, advancing understanding of soliton geometries.

Contribution

It provides the first explicit example of an asymptotically cusped expanding gradient Ricci soliton in three dimensions and proves its uniqueness under certain curvature constraints.

Findings

Constructed a unique asymptotically cusped soliton on R x T^2.

Proved the soliton's uniqueness with negative curvature pinching.

Demonstrated the soliton's asymptotic geometry at both ends.

Abstract

We construct an expanding gradient Ricci soliton in dimension three over the topological manifold R x T^2 (the product of a line and a torus) that aproaches asymptotically a constant curvature cusp at one end, and a flat manifold on the other end. We prove that this is the only gradient soliton with this topology, provided the curvature is negatively pinched, -1/4 < sec < 0, at the time-zero manifold (normalizing the soliton to be born at time -1).