Prescribing almost constant curvatures on manifolds with boundary

TL;DR

This paper studies the existence of conformal metrics on a ball with prescribed scalar and boundary mean curvatures close to constants, extending previous results through a perturbative approach.

Contribution

It introduces new existence results for prescribed curvatures on manifolds with boundary using a perturbative method near constant curvatures.

Findings

Existence of conformal metrics with near-constant prescribed curvatures

Extension of previous boundary curvature problem results

Application of Han and Li's ansatz in a boundary setting

Abstract

In this paper, we investigate a boundary case of the classical prescribed curvature problem. We focus on prescribing the scalar curvature function K and the boundary mean curvature H on the standard ball. Our analysis extendes previous studies by considering the scenario where the curvatures K and H are close to constants. Using a perturbative approach and leveraging the ansatz introduced by Han and Li, we establish new existence results for the conformal metric when the prescribed curvatures are near constants