Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces

TL;DR

This paper demonstrates that the Laplacian and Lichnerowicz Laplacian are sectorial operators on asymptotically hyperbolic spaces, enabling the use of analytic semigroup theory for solving related parabolic equations.

Contribution

It establishes the sectoriality of key Laplacians on asymptotically hyperbolic spaces, facilitating advanced analysis of parabolic PDEs in these geometries.

Findings

Laplacian on functions is sectorial in weighted Hölder spaces.

Lichnerowicz Laplacian on symmetric 2-tensors is sectorial.

Analytic semigroup theory applies to these operators, ensuring well-posedness of parabolic equations.

Abstract

We prove that both the Laplacian on functions, and the Lichnerowicz Laplacian on symmetric 2-tensors with respect to asymptotically hyperbolic metrics, are sectorial maps in weighted H\"older spaces. As an application, the machinery of analytic semigroups then applies to yield well-posedness results for parabolic evolution equations in these spaces.