Imersões mínimas e estimativas do primeiro autovalor do Laplaciano

Abstract

In this dissertation we will present an estimate for the first eigenvalue of the Laplacian of a compact minimal hypersurface Mn of the sphere Sn+1 obtained by Barros and Bessa, we will calculate the Laplacian of the square of the norm of the second fundamental form of an immersion with some applications to estimates of the first eigenvalue of the Laplacian of minimal immersions and we will show two results due to Leung (1983), which estimate the Ricci curvature of a minimal submanifold of the sphere in terms of the second fundamental form of the immersion. The structure of this dissertation is as follows: Chapter 2 will develop all the material inherent to our work, including some identities that relate the Laplacian of an infinite class C function to Mn; Chapter 3 will demonstrate Bochner's and Reilly's formulas, and will also briefly discuss the structure equations on a Riemannaian manifold, aiming to calculate the Laplacian of a symmetric tensor that formally satisfies the Codazzi equations; due to the importance of estimating the first eigenvalue of a Riemannaian manifold, Chapter 4 will apply these formulas, presenting some estimates for lambida1(M). Furthermore, we will demonstrate the result of A. Barros and G. P. Bessa; Chapter 5 will demonstrate the results of P. F. Leung; and finally, Chapter 6 will present some examples of minimal immersions of the Euclidean sphere.