  Lipid membranes constitute very particular materials: on the one hand, they
break very easily under microscopical stretching; on the other hand, they are
extremely flexible, presenting deformations even at small scales. Consequently,
a piece of membrane has an area excess relative to its optically resolvable
area, also called projected area. From a mechanical point of view, we can thus
identify three tensions associated to lipid membranes: the mechanical effective
tension $\tau$, associated to an increase in the projected area and to the
flattening of the fluctuations; the tension $\sigma$, associated to the
microscopical area of the membrane and thus non measurable, but commonly used
in theoretical predictions; and its macroscopical counterpart measured through
the fluctuation spectrum, $r$. Up to now, the equality between these quantities
was taken for granted when analyzing experimental data. In this dissertation,
we have studied, using the projected stress tensor, whether and under which
conditions it is justified to assume $\tau = \sigma$. We studied three
geometries (planar, spherical and cylindrical) and obtained the relation $\tau
\approx \sigma - \sigma_0$, where $\sigma_0$ is a constant depending only on
the membrane's high frequency cutoff and on the temperature. Accordingly, we
conclude that neglecting the difference between $\tau$ and $\sigma$ is
justifiable only to membranes under large tensions: in the case of small
tensions, corrections must be taken into account. We have studied the
implications of this result to the interpretation of experiments involving
membrane nanotubes. Regarding $r$, we have questioned a former demonstration
concerning its equality with $\tau$. Finally, the force fluctuation for planar
membranes and membrane nanotubes was quantified for the first time.
