Abstract. Quadratic curvature Gauss-Bonnet gravity may be the solution to the dark energy problem, but a large coupling strength is required. This can lead to conflict with laboratory and planetary tests of Newton’s law, as well as light bending. The corresponding constraints are derived. If applied directly to cosmological scales, the resulting bound on the density fraction is |ΩGB| . 3.6×10−32.

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

We analytically derive the covariant form of Riemann (curvature) tensor for homogeneous metric-affine cosmologies. That is, we present, in a cosmological setting, most general full including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so compute list curvature by-products such as Ricci tensor, homothetic curvature, scalar, Einstein etc...

The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on R is described in terms of the moduli space parameter k. We find that the distribution of Gaussian curvature on the surface reflects the monopole structure. This is elucidated by the behaviour of the Gauss maps of the minimal surfaces. 2000 Mathematics Subject Classification. Primary 53A10; Secondary 53C07, 81T13 §

A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established. Applications are given to the metric structure of graded ideals in C[z1, . . . , zd], and the existence of “inner” sequences for closed submodules of the free Hilbert module H(C).

In this paper we study the Dirichlet problem for some Monge-Ampère type equations on S, which naturally arise in some geometric problems. The result then is applied to prove the existence of hypersurfaces in R of prescribed Gauss-Kronecker curvature and with fixed boundary.

We describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves, and surfaces of constant negative Gauss curvature. To appear in Journal of Physics A: Mathematical and General PACS numbers: 03.40.Gc, 02.40.+m, 11.10.Lm, 68.10-m 2 RON PERLINE

Is the space of initial data for the Einstein vacuum equations connected? As a partial answer to this question, we prove the following result: Let M be the space of asymptotically flat metrics of non-negative scalar curvature on R3 which admit a global foliation outside a point by 2-spheres of positive mean and Gauss curvatures. Then M is connected.

We present two comparison principles for viscosity suband supersolutions of Monge-Ampère-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.

We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, SchoenSimon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman’s result for the Bernstein type theorem.