  The Gaussian (saddle splay) rigidity of fluid membranes controls their
equilibrium topology but is notoriously difficult to measure. In lipid
mixtures, typical of living cells, linear interfaces separate liquid ordered
(LO) from liquid disordered (LD) bilayer phases at subcritical temperatures.
Here we consider such membranes supported by curved supports that thereby
control the membrane curvatures. We show how spectral analysis of the
fluctuations of the LO-LD interface provides a novel way of measuring the
difference in Gaussian rigidity between the two phases. We provide a number of
conditions for such interface fluctuations to be both experimentally measurable
and sufficiently sensitive to the value of the Gaussian rigidity, whilst
remaining in the perturbative regime of our analysis.

Soft Condensed Matter