Estimativas de volume de subvariedades mínimas compactas imersas em alguns espaços simétricos de posto 1

Abstract

The theory of minimal surfaces came up with a problem proposed by Lagrange, which consisted of the following: given a closed curve without auto-intersections, find the surface of the smallest area that has that curve as a 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 minimums. 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 estimating 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 estimates of volume of minimal compact submanifolds immersed in certain symmetrical spaces of post 1, namely: the unitary sphere Sn , and the real Rpn , complex Cpn and quaternionic Hpn projective spaces. It will be shown that if Mm is a minimal submanifold of the Sn sphere, then vol M ≥ Vc(n, M) where Vc(n, M) is o n-volume according to M. Another estimate for this is vol M ≥ cn, where cn = vol Sn . In the case of M being immersed in projective spaces, we have the lower limits: cn/2 in Rpn , cn+1/2π in Cpn e cn+2/2π 2 in Hpn.