Phase transition for Loewner evolutions with complex linear drivers

We study deterministic Loewner evolutions on the complex plane driven by complex-valued functions. This model can be viewed as a generalization of real-driven Loewner evolutions in the upper half-plane, or as the deterministic analogue of complex-driven Schramm-Loewner evolutions. First, we contribute to the already known theory of such evolutions. We establish a sufficient condition for drivers in the $C^1$-class to create a two-sided simple curve. By constructing a counterexample in the $C^0$-class, we demonstrate that the same condition is not necessary and discuss an alternative necessary and sufficient condition for $C^0$-drivers that create two-sided curves. Second, we analyze the evolutions driven by the one-parameter family of complex linear drivers $\{ct\}_{c \in \mathbb C}$. We show that the geometries of the generated hulls differ significantly from the chordal real-driven case. Although each complex linear driver creates a two-sided curve, the geometry of the generated curve exhibits three distinct geometric phases depending on the complex parameter $c$: a simple phase, a simple with one end spiraling phase, and a third new exotic variant. In this exotic phase, one part of the curve is simple while the other part forms a Jordan curve rooted at the origin. After forming the Jordan curve, this part ceases to grow while disconnecting an open set of positive area from infinity for arbitrarily large times. We determine the phase boundaries in terms of $c$ via the signs of an explicit expression. Within the H\"older-$1/2$-class, we improve the upper bound on a constant sufficient to ensure that the driver creates a two-sided simple curve.