On the characteristic curves on a smooth surface

Abstract

We study some symmetries between two classical pairs of foliations defined on smooth surfaces in ℝ3: the asymptotic curves and the characteristic curves. The asymptotic curves exist in hyperbolic regions of surfaces and have been well studied. The characteristic curves are the analogy of the asymptotic curves in elliptic regions. We produce results on the characteristic curves mirroring those in Uribe‐Vargas (Mosc. Math. J. 6 (2006) 731–768) on the asymptotic curves. By considering cross‐ratios of Legendrian lines in the manifold of contact elements to the surface, we show that certain properties of the characteristic curves are invariant under projective transformations. We establish an analogy of the Beltrami–Enepper theorem. We exhibit the existence of a curve of points of zero torsion of the characteristic curves and determine its behaviour near cusps of Gauss and umbilic points.

Non-polar (11-20) InGaN quantum dots with short exciton lifetimes grown by metal-organic vapor phase epitaxy

“…This is the same mechanism in the formation that was proposed for dot formation employing an N 2 anneal on the c-plane 14 .…”

“…Room temperature CL studies of the annealed epilayer reveal a broad emission band between 410 and 450 nm originating from the strip network, indicating that the strips contain InGaN. An interlinking network of InGaN strips is also seen in addition to the small nanostructures when c-plane InGaN epilayers are annealed in nitrogen, but in that case the network is coarser and more disordered 14 .…”

“…1(a) and (b) are metallic Indium/Gallium droplets formed by the decomposition of the InGaN epilayer, which has been enhanced by the absence of active nitrogen from ammonia gas during the annealing process. The formation mechanism of the nanostructures appears to be analogous to the modified droplet epitaxy that has been used to obtain c-plane InGaN QDs 14 . We investigated the luminescence properties of individual bright spots by taking a spatial CL intensity map of an area consisting of 33 x 22 pixels with a step size of 180 nm [ Fig.…”

“…Growth of InGaN QDs along non-and semi-polar orientations is thus of interest since the internal electric fields can be greatly reduced or eliminated 6,7 . Although there are a few reports on the growth of InGaN nanostructures on non-polar m-plane (1-100) by metal-organic vapor phase epitaxy (MOVPE) 8 and (11-22) semi-polar orientation by molecular beam epitaxy 9 , little progress has been reported on the growth of non-polar a-plane (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) InGaN QDs.…”

Growth of non-polar (11-20) InGaN quantum dots by metal organic vapour phase epitaxy using a two temperature method

“…Non-polar (11)(12)(13)(14)(15)(16)(17)(18)(19)(20) InGaN quantum dots (QDs) were grown by metal organic vapour phase epitaxy. An InGaN epilayer was grown and subjected to a temperature ramp in a nitrogen and ammonia environment before the growth of the GaN capping layer.…”