On the spectrum of minimal submanifolds in space forms

Thursday, May 18, 2017
Luciano Mari
Scuola Normale Superiore

Let $\varphi : M^m \to N^n$ 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
\[
\Theta(r) =\frac{Vol(M\cap B^n_r)}{Vol(\mathbb{B}^m_r )}
\]
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.