Fix constants χ > 0 and θ ∈ [0, 2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e i(h/χ +θ) starting at a fixed boundary point of the domain. Letting θ vary, one obtains a family of curves that look locally like SLE κ processes with κ ∈ (0, 4) (where2 ), which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE 16/κ ) within the same geometry using ordered "light cones" of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE κ (ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE 16/κ (ρ) processes. Mathematics Subject Classification 60J67B Jason Miller

We establish existence and uniqueness for Gaussian free field flow lines started at interior points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at boundary points and use Gaussian free field machinery to determine which chordal SLE κ (ρ 1 ; ρ 2 ) processes are time-reversible when κ < 8. Here we extend these results to wholeplane SLE κ (ρ) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ ∈ [0, 4]) to all κ ∈ [0, 8]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE κ for some κ ∈ (0, 4), and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of SLE κ where κ := 16/κ ∈ (4, ∞). By varying the boundary data we obtain, for each κ > 4, a family of space-filling variants of SLE κ (ρ) whose time reversals belong to the same family. When κ ≥ 8, ordinary SLE κ belongs to this family, and our result shows that its time-reversal is SLE κ (κ /2−4; κ /2−4). As applications of this theory, we obtain the local finiteness of CLE κ , for κ ∈ (4, 8), and describe the laws of the boundaries of SLE κ processes stopped at stopping times. Mathematics Subject Classification 60J67B Jason Miller