  Theoretical studies of nearly spherical vesicles and microemulsion droplets,
that present typical examples for thermally-excited systems that are subject to
constraints, are reviewed. We consider the shape fluctuations of such systems
constrained by fixed area $A$ and fixed volume $V$, whose geometry is presented
in terms of scalar spherical harmonics. These constraints can be incorporated
in the theory in different ways. After an introductory review of the two
approaches: with an exactly fixed by delta-function membrane area $A$ [Seifert,
Z. Phys. B, 97, 299, (1995)] or approximatively by means of a Lagrange
multiplier $\sigma$ conjugated to $A$ [Milner and Safran, Phys. Rev. A, 36,
4371 (1987)], we discuss the determined role of the stretching effects, that
has been announced in the framework of a model containing stretching energy
term, expressed via the membrane vesicle tension [Bivas and Tonchev,
Phys.Rev.E, 100, 022416 (2019)]. Since the fluctuation spectrum for the used
Hamiltonian is not exactly solvable an approximating method based on the
Bogoliubov inequalities for the free energy has been developed. The area
constraint in the last approach appears as a self-consistent equation for the
membrane tension. In the general case this equation is intractable
analytically. However, much insight into the physics behind can be obtained
either imposing some restrictions on the values of the model parameters, or
studying limiting cases, in which the self-consistent equation is solved.
Implications for the equivalence of ensembles have been discussed as well.
