相关论文: About the integrability of the Rapcs\'ak equation
In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used…
Several complete systems of integrability conditions on a spin chain Hamiltonian density matrix are presented. The corresponding formulas for $R$-matrices are also given. The latter is expressed via the local Hamiltonian density in the form…
We describe the Special K\"ahler structure on the base of the so-called Hitchin system in terms of the geometry of the space of spectral curves. It yields a simple formula for the K\"ahler potential. This extends to the case of a singular…
We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.
n this paper we formulate necessary conditions for the integrability in the Jacobi sense of Newton equations $\ddot \vq=-\vF(\vq)$, where $\vq\in\C^n$ and all components of $\vF$ are polynomial and homogeneous of the same degree $l$. These…
We study two integrable systems associated with the coupled NLS equation: the integrable defect system and the integrable boundary systems. Regarding the first one, we present a type I defect condition, which is described by a B\"{a}cklund…
A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if…
In this paper we give sufficient conditions on a compact orbifold with an extremal Kaehler metric to admit a resolution with an extremal Kaehler metric. We also complete the Kaehler constant scalar curvature case.
First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective…
We study the existence of extremal K\"ahler metrics on K\"ahler manifolds. After introducing a notion of relative K-stability for K\"ahler manifolds, we prove that K\"ahler manifolds admitting extremal K\"ahler metrics are relatively…
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the…
It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy…
We consider a natural Hamiltonian system of $n$ degrees of freedom with a homogeneous potential. Such system is called partially integrable if it admits $1<l<n$ independent and commuting first integrals, and it is called super-integrable if…
A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions. Existence and uniqueness of the solution is proved by…
In complex Finsler geometry, an open problem is: does there exist a weakly K\"ahler Finsler metric which is not K\"ahler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly…
The second author has shown that existence of extremal K\"ahler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to…
Methods developed for the analysis of integrable systems are used to study the problem of hyperK\"ahler metrics building as formulated in D=2 N=4 supersymmetric harmonic superspace. We show, in particular, that the constraint equation…
We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior $C^{1, \gamma}$…
We treat Koll\'ar's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Koll\'ar type cohomology injectivity theorems. Our main theorem is formulated for a…