Trichotomy Theorem for Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundaries

TL;DR

This paper extends the Trichotomy Theorem to prescribe scalar and mean curvatures on compact manifolds with boundary, classifying possibilities based on the conformal Laplacian's eigenvalues, and discusses related curvature problems on Riemann surfaces.

Contribution

It introduces a comprehensive Trichotomy Theorem for scalar and mean curvature prescription on manifolds with boundary, generalizing previous results and including a new iteration scheme for boundary conditions.

Findings

Classification of curvature prescription possibilities based on eigenvalues

Extension of the Trichotomy Theorem to manifolds with boundary

Development of a monotone iteration scheme for boundary nonlinearities

Abstract

In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact Riemann surfaces with boundary for metrics either pointwise conformal or conformally equivalent to the original metric, provided that the Euler characteristic is negative. The key step is a general version of monotone iteration scheme which handle the zeroth order nonlinear term on the boundary conditions.