Kazdan-Warner equation on graph

The Lp-Minkowski Problem and the Minkowski Problem in Centroaffine Geometry

The Lp-Minkowski problem introduced by Lutwak is solved for p > n + 1 in the smooth category. The relevant Monge-Ampère equation (1) is solved for all p > 1. The same equation for p < 1 is also studied and solved for p ∈ (−n− 1, 1). When p = −n− 1 the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner type obstruction for this problem is obtained. ∗kschou@m...

Annals of Mathematics Curvature Functions for Compact 2 - Manifolds

An Obstruction for the Mean Curvature of a Conformal

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S → R satisfies ∫ ∂XH = 0 where X is a conformal vector field on S and where the integration is carried out with respect to the Euclidean volume measure of the image. This identity ...

The Scalar Curvature Equation on S 3

An obvious necessary condition for the existence of solutions to (1.1) is that the function K has to be positive somewhere. Moreover, there are the Kazdan-Warner obstructions [7, 16], which imply in particular, that a monotone function of the coordinate function X1 cannot be realized as the scalar curvature of a metric conformal to g0. Numerous studies have been made on equation (1.1) and its h...

Existence and Uniqueness of Constant Mean Curvature Foliation of Asymptotically Hyperbolic 3-manifolds Ii

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a th...