  Pattern-forming processes, such as electrodeposition, dielectric breakdown or
viscous fingering are often driven by instabilities. Accordingly, the resulting
growth patterns are usually highly branched, fractal structures. However, in
some of the unstable growth processes the envelope of the structure grows in a
highly regular manner, with the perturbations smoothed out over the course of
time. In this paper, we show that the regularity of the envelope growth can be
connected to small-scale instabilities leading to the tip splitting of the
fingers at the advancing front of the structure. Whenever the growth velocity
becomes too large, the finger splits into two branches. In this way it can
absorb an increased flux and thus damp the instability. Hence, somewhat
counterintuitively, the instability at a small scale results in a stability at
a larger scale. The quantitative analysis of these effects is provided by means
of the Loewner equation, which one can use to reduce the problem of the
interface motion to that of the evolution of the conformal mapping onto the
complex plane. This allows an effective analysis of the multifingered growth in
a variety of different geometries. We show how the geometry impacts the shape
of the envelope of the growing pattern and compare the results with those
observed in natural systems.
