A Functional Connected With a Surface of Given Mean Curvature

Abstract

Microwave power, P ECH < 140 kW, has been injected at 28 GHz into the axisymmetric plug/barrier cell in the axisymmetrized tandem mirror GAMMA 10. As observed by soft X-ray measurements, the microwaves generate a hot (50-60 keV) electron population, radially peaked on the magnetic axis, which results in the formation of a thermal barrier. The production mechanism of these hot electrons is found to be second harmonic electron cyclotron heating (ECH), corrected for the effects of the relativistic mass variation and the Doppler shift. This mechanism also explains the first experimental observation of a saturation of the single-component hot electron temperature T eh as being caused by the finite width of the incident microwave lobe. The dependence of the plasma parameters on the filling gas pressure, the plasma density and the ECH power is studied. It is found that the heating process can be interpreted as a competition between electron acceleration by the incident wave, electron deceleration by collisions, and the mirror trapping efficiency of the source electrons for hot electrons. The axial profile of the soft X-rays is investigated in relation to the mechanism of the second harmonic ECH. The heating process is discussed in terms of the electron pitch angle and the magnetic field intensity.

The small sphere limit of quasilocal energy in higher dimensions along lightcone cuts

“…One important feature that distinguishes this approach from others is that it requires a flat reference via isometric embedding of the codimension-2 surface to the Minkowski spacetime as the zero-point energy. In four dimensions, the existence of the isometric embedding is guaranteed by the results by Nirenberg [23] and Pogorelov [24]. In particular, we will be considering the lightcone reference, where S is embedded on a lightcone in the Minkowski spacetime.…”

Prescribed curvature measure problem in hyperbolic space

“…The general Christoffel-Minkowski problem is a problem of prescribing a given k-th area measure. There is a vast literature devoted to the study of these types of problems, and we refer to [1,4,8,10,12,21,22,24,26], and [13] and the references therein.…”