18 may 2017 -- 16:00
Sala Riunioni, Dipartimento di Matematica, Pisa
Abstract.
Let $\varphi : M^m → N^n$ be an immersed minimal submanifold in Euclidean or hyperbolic space. In this talk, I survey on some recent results obtained in collaboration with various colleagues from Brazil, to ensure that the Laplace-Beltrami operator of $M$ has purely discrete (respectively, purely essential) spectrum. In the last case, we also give an explicit description of the spectrum. Our criteria apply to many examples of minimal submanifolds constructed in the literature, and answer a question posed by S.T.Yau. The geometric conditions involve the Hausdorff dimension of the limit set of $\varphi$ and the behaviour at infinity of the density function $ \Theta(r) =\frac{\text{vol}(M\cap B^n_r)}{\text{vol}(\mathbb{B}^m_r )} $ where $B^n_r,\mathbb{B}^m_r$ are geodesic balls of radius $r$ in $N^n$ and $N^m$, respectively. This is based on joint works with G.Pacelli Bessa, L.P. Jorge, J.F. Montenegro, B.P. Lima, F.B. Vieira.