arXiv subjects

Petros Siasos

Publications and source records attributed to Petros Siasos.

2 recordsLinked to original sources

arXiv2026

The Heat Kernel of the Hodge Laplacian on an Invariant Subspace of 1-forms on the Hyperbolic Space $\mathbb{H}^{n}$

In this paper, we construct explicit formulas for the heat kernel of the Hodge Laplacian restricted to an invariant subspace of $1$-forms on the real hyperbolic space $\mathbb{H}^{n}$. Using warped product decompositions, the problem is reduced to component operators on specific subspaces, yielding precise closed-form expressions for both odd and even dimensions. We further establish Millson-type recurrence relations that connect heat kernels across differing dimensions. Based on these explicit representations, we derive sharp uniform estimates and long-time asymptotic behavior, extending classical scalar heat kernel bounds to $1$-forms on real hyperbolic spaces.

math.DG↗

arXiv2024

The $L^p$-spectrum of the Laplacian on forms over warped products and Kleinian groups

In this article, we generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we prove that the $L^p$-spectrum of the Laplacian on $k$-forms contains a parabolic region which depends on $k$, $p$ and the limiting curvature $a_0$ at infinity. For $M=\mathbb{H}^{N+1}/Γ$ with $Γ$ a geometrically finite group such that $M$ has infinite volume and no cusps, we prove that the $L^p$-spectrum of the Laplacian on $k$-forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line.

math.DG↗