We address the asymptotic behavior of ?-Gauss curvature flow, for ?>1/2, with a complete non-compact convex initial hypersurface which is contained in cylinder bounded cross section. show that flow converges, as t?+?, locally smoothly to translating soliton uniquely determined by hypersurface.

We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map, a canonical transform between the ADM reduced variables and holonomy variables can be constructed. This allows one to solve (in principle) for the evolution...

Let S be a complete, open simply connected surface. Suppose that the integral of the Gauss curvature over arbitrary measurable sets is less than tt/2 in magnitude. We show that the surface admits a global Tchebychev net.

We show that the Gauss-Bonnet term is the only consistent curvature squared interaction in the Randall-Sundrum model and various static and inflationary solutions can be found. And from metric perturbations around the RS background with a single brane embedded, we also show that for a vanishing Gauss-Bonnet coefficient, the brane bending allows us to reproduce the 4D Einstein gravity at the lin...

We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with non-trivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action i...