  The biological function of membranes is closely related to their softness,
which is often studied through the membranes' thermally-driven fluctuations.
The analysis commonly assumes that the relaxation rate of a pure bending
deformation is determined by the competition between membrane bending rigidity
and viscous dissipation in the surrounding medium. Here, we reexamine this
assumption and demonstrate that viscous flows within the membrane dominate the
dynamics of bending fluctuations of non-planar membranes with a radius of
curvature smaller than the Saffman-Delbr\"uck length. Using flickering
spectroscopy of giant vesicles made of DPPC:Cholesterol mixtures and pure
diblock-copolymer membranes, we experimentally detect the signature of membrane
dissipation in curvature fluctuations, and show that membrane viscosity can be
reliably obtained from the short time behavior of the shape time correlations.
The results indicate that the DPPC:Cholesterol membranes behave as a Newtonian
fluid, while polymer membranes exhibit more complex rheology. Our study
provides physical insights into the time scales of curvature remodeling of
biological and synthetic membranes.
