The theory of minimal surfaces came up with a problem proposed by Lagrange, which consisted
of the following: given a closed curve without self-intersections, find the surface of
smallest area that has that curve as boundary. Such a problem became known as
the Plateau Problem. It took about 16 years from Lagrange's work to discover non-trivial
examples of minimal surfaces due to Meusnier. The theory was stagnant for 60 years until
Scherk found new examples of minimal surfaces. With the work of Weierstrass it was possible to
obtain more examples of these surfaces. Thereafter there were major developments in theory,
becoming one of the most fertile fields of Differential Geometry. One class of problems studied
is that of estimate the volume of minimal submanifolds immersed in certain ambient
manifolds, such as spheres, hyperplanes, projective spaces, etc. The objective of the present
work is to provide lower bounds of volume of minimal compact submanifolds immersed
in certain symmetrical spaces of rank 1, namely: the unitary sphere Sn, and the real projective space RPn,
complex projective space CPn and quaternionic projective space HPn. It will be shown that if Mm is a
minimal submanifold of Sn, then volM >=V_c(n;M) where V_c(n;M) is the n-conformal volume
of M. Another estimate for this is volM>=c_n, where c_n = vol(Sn). In the case of
M being immersed in projective spaces, we have the lower bounds: c_n/2 in RPm, c_{n+1}/2\pi in
CPm and c_{n+2}=2\pi^2 in HPm.