A class of Neumann problems arising in conformal geometry

Abstract

In this paper, we solve a class of Neumann problems on a manifold with totally geodesic smooth boundary. As a consequence, we also solve the prescribing k-curvature problem of the modified Schouten tensor on such manifolds; that is, if the initial k-curvature of the modified Schouten tensor is positive for τ > n − 1 or negative for τ < 1, then there exists a conformal metric such that its k-curvature defined by the modified Schouten tensor equals some prescribed function and the boundary remains totally geodesic.

The Neumann problem for a type of fully nonlinear complex equations

“…Comparing to the complex fully nonlinear equations, the works about the real fully nonlinear equations are more abundant. We refer to [8,24,23,9,17,18,32,33,22,21,4,5,12,13,16] and references therein. We remark that in the above mentioned papers, the domain is assumed to be strictly pseudoconvex, convex or mean convex.…”

A class of fully nonlinear equations arising in conformal geometry

“…if −Ric g 0 ∈ Γ k . For the manifolds with totally geodesic boundary, [ShY14] showed that for some positive function f (x) and h < k there exists a conformal metric g such that…”