Riemannian Metrics on Locally Projectively Flat Manifolds

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential in Kahler geometry. If M is compact and u is a negative section of the dual of the tautological bundle whose Hessian is positive definite, then M is projectively equivalent to a quotient of a bounded convex domain in R^n. The same is true if M has a boundary on which u=0. This theorem is analogous to a result of Schoen and Yau in locally conformally flat geometry. The proof uses affine differential geometry techniques developed by Cheng and Yau.Comment: 16 pages, to be published in American Journal of Mathematic

Minimal Lagrangian surfaces in CH^2 and representations of surface groups into SU(2,1)

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in the complex hyperbolic plane CH^2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus recovering a result of Guichard and Parker-Platis. Our proof involves using the Toda lattice framework to construct an SU(2,1) frame corresponding to a minimal Lagrangian surface. Then the equation of Tzitzeica type is an integrability condition. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck

THE STRUCTURE OF THE FOOD SERVICE SECTOR: THE SYSCO EXPERIENCE

Industrial Organization,

Wind tunnel airstream oscillating apparatus Patent

Wind tunnel air flow modulating device and apparatus for selectively generating wave motion in wind tunnel airstrea

Cubic Differentials in the Differential Geometry of Surfaces

We discuss the local differential geometry of convex affine spheres in \re^3 and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for $\mathfrak a^{(2)}_2$. We also discuss the global theory and applications to representations of surface groups and to mirror symmetry.Comment: corrected published editio