One of the central questions in Geometric Analysis is the
interplay between the curvature of the manifold and the spectrum
of an operator.

In this talk, we will be considering the Hodge Laplacian on
differential forms of any order $k$ in the Banach Space $L^p$. In
particular, under sufficient curvature conditions, it will be
demonstrated that the $L^p\,$ spectrum is independent of $p$ for
$1\!\leq\!p\!\leq\! \infty.$ The underlying space is a
$C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound
on its Ricci Curvature and the Weitzenb\"ock Tensor. The further
assumption on subexponential growth of the manifold is also
necessary. We will see that in the case of Hyperbolic space the
$L^p$ spectrum does in fact depend on $p.$

As an application, we will show that the spectrum of the Laplacian
on one-forms has no gaps on certain manifolds with a pole and on
manifolds that are in a warped product form. This will be done
under weaker curvature restrictions than what have been used
previously; it will be achieved by finding the $L^1$ spectrum of
the Laplacian.

Time permitting, we will take a short look at an alternative
method for finding the Gaussian Heat kernel bounds for the Hodge
Laplacian via Logarithmic Sobolev Inequalities. Such bounds are
necessary in the proof of the $L^p$ independence.