Heat-type equations on manifolds with fibered boundary I: Schauder estimates

TL;DR

This paper establishes Schauder estimates for parabolic equations involving the Laplace-Beltrami operator on manifolds with fibered boundaries, extending analysis tools to complex geometric settings relevant for geometric flows.

Contribution

It provides the first parabolic Schauder estimates in the fibered boundary setting with $ ext{Phi}$-metrics, broadening the scope of geometric analysis on such manifolds.

Findings

Proved Schauder estimates for the Laplace-Beltrami operator on fibered boundary manifolds.

Extended analysis techniques to include asymptotically conical and monopole spaces.

Laid groundwork for studying geometric flows like Yamabe and mean curvature flows in this setting.

Abstract

In this paper we prove parabolic Schauder estimates for the Laplace-Beltrami operator on a manifold $M$ with fibered boundary and a $\Phi$-metric $g_\Phi$. This setting generalizes the asymptotically conical (scattering) spaces and includes special cases of magnetic and gravitational monopoles. This paper, combined with part II, lay the crucial groundwork for forthcoming discussions on geometric flows in this setting; especially the Yamabe- and mean curvature flow.