Cota inferior para autovalores de hipersuperfícies mínimas mergulhadas na esfera euclidiana

• Ano de defesa: 2011
• Autor(a) principal: Veras, Tiago Mendonça Lucena de
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Palavras-chave em Português: Variedades riemanianas; Autovalores; Geometria diferencial
• Link de acesso: http://www.repositorio.ufc.br/handle/riufc/1148

Let M be a closed oriented Riemannian manifold and x : Mn → Sn+1 С Rn+2 a minimal immersion of Mn in the Euclidean unit sphere. We know by Takahashi’s theorem Δx + nx=0, where x (p) = (x1 (p ),..., xn +2 (p)) and Δx (p) = (Δx1 (p), ... , Δxn +2 (p)) where Δ denotes the Laplacian on M the induced metric for x, see [11]. It follows that n is an upper bound for the first eigenvalue λ1 of Δ. When x is a embedded in 1982 was conjectured by Yau in [12] that the first eigenvalue of the Laplacian, denoted by λ1, is equal n. The first global result in the direction of such problem was obtained by Choi and Wang in cite Choi where it was proved that λ1 ≥ n / 2. In the article [2] Barros and Bessa showed that λ1 ≥ n / 2 + С (Mn, x), where С (Mn, x) is a positive constant which depends on Mn and x. The aim of this work is to present some conditions for the first eigenvalue of the Laplacian is equal to n, in other words, Yau's conjecture is true under these conditions.