The Indefinite Self-Dual Metrics and Painleve Equations

The Scalar-flat Kaehler Metric and Painleve III

We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.

第六类Painlevé方程表示空间上的Okamoto对称性

Okamoto's symmetry on the representation space of the sixth Painlevé equation

通过Teichmüller $\mathcal{X}$-坐标化的乘法中间卷积，在簇$\mathcal{X}$-突变语言中给出第六类Painlevé方程的Okamoto对称性$w_2$的单值实现，并用彩色等边三角剖分和星形脂肪图编码突变公式。

Antisymmetric Tensor Fields, 4-Vector Fields, Indefinite Metrics and Normalization

On the basis of our recent modifications of the Dirac formalism we generalize the Bargmann-Wigner formalism for higher spins to be compatible with other formalisms for bosons. Relations with dual electrodynamics, with the Ogievetskii-Polubarinov notoph and the Weinberg 2(2J+1) theory are found. Next, we introduce the dual analogues of the Riemann tensor and derive corresponding dynamical equations in the Minkowski space. Relations with the Marques-Spehler chiral gravity theory are discussed. The problem of indefinite metrics, particularly, in quantization of 4-vector fields is clarified.

Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3. An interesting sub-class of these conformal structures are represen…

Dual Isomonodromic Deformations and Moment Maps to Loop Algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix different…

Point Equivalence of Second-Order ODEs: Maximal Invariant Classification Order

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlevé-I equation can be characterized as the simplest second-order ODE belonging to the class of equations requiring 10th order jets for their classification.