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 when the manifold is locally conformally flat or the Ricci curvature is positive.     

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.     

Regular Submanifolds in Conformal Space ${\mathbb Q}^n_p$

The authors study the regular submanifolds in the conformal space Q_p~n and introduce the submanifold theory in the conformal space Q_p~n.The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal spaceQ_p~n is given.Finally,the conformal isotropic submanifolds in the conformal space Q_p~n are classified.     

On the decomposition of global conformal invariants II

This paper is a continuation of [S. Alexakis, The decomposition of global conformal invariants I, submitted for publication, see also math.DG/0509571], where we complete our partial proof of the Deser-Schwimmer conjecture on the structure of “global conformal invariants.” Our theorem deals with such invariants P(gn) that locally depend only on the curvature tensor Rijkl (without covariant derivatives).In [S. Alexakis, The decomposition of global conformal invariants I, Ann. of Math., in press] we developed a powerful tool, the “super divergence formula” which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator Ign(?) that measures the “non-conformally invariant part” of P(gn). This paper resolves the problem of using this information we have obtained on the structure of Ign(?) to understand the structure of P(gn).     

Time-Like Conformal Homogeneous Hypersurfaces with Three Distinct Principal Curvatures

A hypersurface x(M) in Lorentzian space R_1~4 is called conformal homogeneous,if for any two points p, q on M, there exists σ, a conformal transformation of R_1~4, such thatσ(x(M)) = x(M), σ(x(p)) = x(q). In this paper, the authors give a complete classification for regular time-like conformal homogeneous hypersurfaces in R_1~4 with three distinct principal curvatures.     

Conformal mapping and ellipses

We answer a question raised by M. Chuaqui, P. Duren, and B. Osgood by showing that a conformal mapping of a simply connected domain cannot take two circles onto two proper ellipses.

     

Conformal vector fields on spacetimes

We study conformal vector fields and their zeros on spacetimes which are non-conformally-flat. Depending on the Petrov type, we classify all conformal vector fields with zeros. The problems of reducing a conformal vector field to a homothetic vector field are considered. We show that a spacetime admitting a proper homothetic vector field is (locally) a plane wave. This precises a well-known theorem of {Alekseevski}, where all these spacetimes are determined in a more general form.     

A mesh simplification strategy for a spatial regression analysis over the cortical surface of the brain

We present a new mesh simplification technique developed for a statistical analysis of a large data set distributed on a generic complex surface, topologically equivalent to a sphere. In particular, we focus on an application to cortical surface thickness data. The aim of this approach is to produce a simplified mesh which does not distort the original data distribution so that the statistical estimates computed over the new mesh exhibit good inferential properties. To do this, we propose an iterative technique that, for each iteration, contracts the edge of the mesh with the lowest value of a cost function. This cost function takes into account both the geometry of the surface and the distribution of the data on it. After the data are associated with the simplified mesh, they are analyzed via a spatial regression model for non-planar domains. In particular, we resort to a penalized regression method that first conformally maps the simplified cortical surface mesh into a planar region. Then, existing planar spatial smoothing techniques are extended to non-planar domains by suitably including the flattening phase. The effectiveness of the entire process is numerically demonstrated via a simulation study and an application to cortical surface thickness data.     

Conformal infinitesimal variations of submanifolds

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds are a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

秩为 2 的 Virasoro-型李共形代数

作者对秩为2的无挠的李共形代数进行了刻画.在这些代数中,作者主要关注Virasoro-型李共形代数.并且,作者描述了一种特殊Virasoro-型李共形代数的共形导子、秩为1的自由共形模和中心扩张.     

Asymptotics of the Best Polynomial Approximation of?|x|p and?of?the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| ^p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.      

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

On the curvature conjecture of Hua Loo-keng

It is proved that both the holomorphic sectional and the bisectional curvatures of the conformal Bergman metric ds21 = K2(z,)2log K(z, )/zαβdzαdβ are always negative, where K(z,) is the Bergman kernel of a bounded domain Din Cn . As a subsequent result, the Weyl tensor for a Hermitian manifold is obtained.     

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm-Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm-Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm-Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm-Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.     

Preservation of local dynamics when applying central difference methods: application to SIR model

We apply the central difference method (u _t+1 ? u _t ? 1)/(2Δt) = f(u _t ) to an epidemic SIR model and show how the local stability of the equilibria is changed after applying the numerical method. The above central difference scheme can be used as a numerical method to produce a discrete-time model that possesses interesting local dynamics which appears inconsistent with the continuous model. Any fixed point of a differential equation will become an unstable saddle node after applying this method. Two other implicitly defined central difference methods are also discussed here. These two methods are more efficient for preserving the local stability of the fixed points for the continuous models. We apply conformal mapping theory in complex analysis to verify the local stability results.     

负曲率流形上给定数量曲率的共形形变

本文研究具强负曲率Cartan－Hadamard流形M~n（n≥3）上给定数量曲率函数S的共形形变问题．利用上下解方法,并通过精心构造上解,我们获得了当完备的共形形变度量存在时,函数S在无穷远附近的最佳渐近性态．在较一般情况下,我们还给出了共形数量曲率方程解的渐近估计．     

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

An Accelerated Osculation Method and its Application to Numerical Conformal Mapping

A variant of the classical Koebe-logarithm osculation algorithm for conformal mapping is obtained by inserting a hyperbolic sine at an intermediate step. The modulus of convergence is calculated, and numerical experiments are reported, in particular in comparison with the method of Grassmann [E. Grassmann (1979). Numerical experiments with a method of successive approximation for conformal mapping. J. Applied Mathematics and Physics, 30, 873-884.]. Either procedure may work better, depending upon the domain. Further numerical examples show how the osculation method can be coupled to faster converging algorithms (which tend to work best for nearly-circular domains), thus making feasible computations which would not be accessible by either method alone.     

On representing some lattices as lattices of intermediate subfactors of finite index

We prove that the very simple lattices which consist of a largest, a smallest and 2n pairwise incomparable elements where n is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite depth. Using the same techniques, we give a necessary and sufficient condition for subfactors coming from Loop groups of type A at generic levels to be maximal.