-valued differential forms on
The Dirac equation for spinor-valued fields on the Minkowski space of even dimension form a hyperbolic system of partial differential equations. In the paper, we are showing how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula expressing the value of in a chosen point as an integral over a compact cycle given by the intersection of the null cone with in the Minkowski space .
In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface....
Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as -modules. As finite-dimensional irreducible -modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
We present necessary and sufficient conditions for a measure to be a p-Carleson measure, based on the Poisson and Poisson-Szegő kernels of the n-dimensional unit ball.