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Nelia Charalambous : On the $L^p$ Spectrum of the Hodge Laplacian on Non-Compact Manifolds



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.

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