Estimando o primeiro autovalor do Laplaciano de hipersuperfícies mínimas

Abstract

In the article A first eigenvalue estimate for minimal hypersurfaces [1] H. Choi e A. N. Wang obtain a lower bound for the first eigenvalue of the Laplacian of a compact orientable embedded minimal hypersurface in an compact orientable manifold with Positive Ricci curvature. In this dissertation, using covering space argument we prove this result dropping the orientability assumption. Moreover, we use Reilly’s Formula was used, which is actually a version obtained by integrating of Bochner’s Formula. Combining Choi and Wang’s estimate with Yang and Yaús Theorem [2], we found a upper bounds estimate for the area of an embedded minimal surface in S 3 only depending on its topology, more precisely only in terms of the genus of the surface.