Rigidity of asymptotically conical shrinking gradient Ricci solitons

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on R^n.Comment: 44 page

The gradient of a graph

In this article I introduce a dynamic interpretation of the gradient of a graph which leads naturally into the notion of differentiation

Second harmonic generation on self-assembled GaAs/Au nanowires with thickness gradient

Here we investigated the SH generation at the wavelength of 400 nm (pump laser at 800 nm, 120 fs pulses) of a "metasurface" composed by an alternation of GaAs nano-grooves and Au nanowires capping portions of flat GaAs. The nano-grooves depth and the Au nanowires thickness gradually vary across the sample. The samples are obtained by ion bombardment at glancing angle on a 150 nm Au mask evaporated on a GaAs plane wafer. The irradiation process erodes anisotropically the surface, creating Au nanowires and, at high ion dose, grooves in the underlying GaAs substrate (pattern transfer). The SHG measurements are performed for different pump linear polarization angle at different positions on the "metasurface" in order to explore the regions with optimal conditions for SHG efficiency. The pump polarization angle is scanned by rotating a half-wave retarder plate. While the output SH signal in reflection is analyzed by setting the polarizer in s or p configuration in front of the detector. The best polarization condition for SHG is obtained in the configuration where the pump and second harmonic fields are both p polarized, and the experiments show a SH polarization dependence of the same symmetry of bulk GaAs. Thus, the presence of gold contributes only as field localization effect, but do not contributes directly as SH generator