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

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.