Spacefilling curves and phases of the Loewner equation Joan LindS. Rohde 30C20Loewner equation Similar to the well-known phases of SLE, the Loewner differential equation with $\operatorname{Lip}(1/2)$ driving terms is known to have a phase transition at norm $4$, when traces change from simple to nonsimple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: there is a constant $C>4$ such that a Loewner driving term whose trace is space filling has $\operatorname{Lip}(1/2)$ norm of at least $C$. We also provide a geometric criterion for traces to be driven by $\operatorname{Lip}(1/2)$ functions, and show how examples such as the Hilbert space-filling curve and the Sierpinski gasket fall into this class. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4794 10.1512/iumj.2012.61.4794 en Indiana Univ. Math. J. 61 (2012) 2231 - 2249 state-of-the-art mathematics http://iumj.org/access/