A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations wb.en the manifold is locally conformally flat or the Ricci curvature is positive.

Nonlinear Conformal Gravitation

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.

De Sitter Space as a Computational Tool for Surfaces and Foliations

The set of all spheres and hyperplanes in the Euclidean space Rn+1 could be identified with the Sitter space Λn+1. All the conformal properties are invariant by the Lorentz form which is natural pseudo-Riemannian metric on Λn+1. We shall study behaviour of some surfaces and foliations as their families using computation in the de Sitter space.

Ab initio investigation of boron nanodevices:conductances of the different geometric conformations

Conductances of different geometric conformations of boron ribbon devices are calculated by the ab initio method, The I-V characteristics of three devices are rather different due to the difference in structure. The current of the hexagonal boron device is the largest and increases nonlinearly. The current of the hybrid hexagon-triangle boron device displays a large low-bias current and saturates at a value of about 5.2 uA, The current of the flat triangular boron flake exhibits a voltage gap at low bias and rises sharply with increasing voltage. The flat triangular boron device can be either conducting or insulating, depending on the field.

Conformal CMC-Surfaces in Lorentzian Space Forms

Let Q^3 be the common conformal compactification space of the Lorentzian space forms Q^3 1 ,S^3 1,We study the conformal geometry of space-like surfaces in Q^3 ,It is shown that any conformal CMC-surface in Q^3 must be conformally equivalent to a constant mean curvature surface in R^3 1,or,H^3 1,We also show that if x ：M→Q^3 is a space-like Willmore surface whose conformal metric g has constant curvature K,the either K = -1 and x is conformally equivalent to a minimal surface in R^3 1,or K=0 and x is conformally equivalent to the surface H^1（1/√2）×H^1（1/√2） in H^3 1.

Conformally flat Lorentzian hypersurfaces in R_1~4 with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x : M_1~3→ R_1~4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of R_1~4. Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.

Completion of R^(2)with a Conformal Metric as a Closed Surface

In this paper,we obtain some asy mptotic behav ior results for solutions to the prescribed Gaussian curvature equation.Moreover,we prove that under a con-formal metric in R^(2),if the total Gaussian curvature is 4π,the conformal area of R^(2)is finite and the Gaussian curvature is bounded,then R^(2)is a compact C^(l,α)surface after completion at∞,for anya∈(0,1).If the Gaussian curvature has a Holder decay at in-finity,then the completed surface is C^(2).For radial solutions,the same regularity holds if the Gaussian curvature has a limit at infinity.