# Geometria em Lisboa Seminar

### On the radial spectrum of rotationally symmetric manifolds

Let $\Omega =B(p,r)$ be a geodesic ball  centered at  $p$ and radius $r>0$ of a Riemannian manifold  $(M,ds^2)$. Let $\sigma (\Omega)=\{\lambda_{k}\}_{k=1}^{\infty}\subset [0, \infty)$ be the spectrum of  $L=-\triangle\vert_{W_{0}^{2}}(\Omega)$.

The radial spectrum of $\Omega$ is the set of  the eigenvalues $\lambda_{k_i}\in \sigma (\Omega)\subset \sigma (\Omega)$ whose eigenfunctions are radial functions,  i.e.,   functions that depend only on the distance $r_p(x)={\rm dist}_{M}(p,x)$.

Recall that  a $m$-dimensional rotationally symmetric manifold with radial sectional curvature $-G(r)$, where $G$ is a smooth even function on $\mathbb{R}$ is defined as the space $\mathbb{M}^m_{h}=[0,R_0)\times\mathbb{S}^{m-1}/\sim$ with ($(t, \theta)\sim (s,\beta)\Leftrightarrow t=s=0$ and $\forall \theta, \beta\in \mathbb{S}^{m-1}$ or $t=s$ and $\theta=\beta$, endowed with the metric $ds^2_{h}(t,\theta)=dt^2+h^2(t)d\theta^2$ where $\sigma$ denotes the unique solution of the Cauchy problem on $[0,R_h)$ \begin{eqnarray}\label{ubs2}
\left\{\begin{array}{l}
h''-Gh=0,\\
h(0)=0, h'(0)=1
\end{array}\right.
\end{eqnarray}and $R_h$ is the largest positive real number such that $h>0$.  Our main result in this talk is the following theorem.

Let $B_r(o)\subset \mathbb{M}_{h}^{n}$ be the geodesic ball of radius $r$ centred at the origin $o$. Let $\sigma^{\rm rad}(B_r(o))=\{\lambda_{1}^{\rm rad}(B_r(o))\leq \lambda_{2}^{\rm rad}(B_r(o)) \leq \cdots\}$ be the radial spectrum of $B_r(o)$. The following identity is true. \label{eqMain}\sum_{i=1}^{\infty}\frac{1}{\lambda_{i}^{\rm
If $\mathbb{M}_{h}^{n}$ is stochastically incomplete  then \label{eqMain2}\sum_{i=1}^{\infty}\frac{1}{\lambda_{i}^{\rm
Here $V(r)={\rm vol}(B_r(o))$ and $S(r)={\rm vol}(\partial B_r(o))$.