Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.Comment: Revised version, to appear on Israel Journal of Mathematics. New title and added section 4 on umbilical surface

NDDepth: Normal-Distance Assisted Monocular Depth Estimation and Completion

Over the past few years, monocular depth estimation and completion have been paid more and more attention from the computer vision community because of their widespread applications. In this paper, we introduce novel physics (geometry)-driven deep learning frameworks for these two tasks by assuming that 3D scenes are constituted with piece-wise planes. Instead of directly estimating the depth map or completing the sparse depth map, we propose to estimate the surface normal and plane-to-origin distance maps or complete the sparse surface normal and distance maps as intermediate outputs. To this end, we develop a normal-distance head that outputs pixel-level surface normal and distance. Meanwhile, the surface normal and distance maps are regularized by a developed plane-aware consistency constraint, which are then transformed into depth maps. Furthermore, we integrate an additional depth head to strengthen the robustness of the proposed frameworks. Extensive experiments on the NYU-Depth-v2, KITTI and SUN RGB-D datasets demonstrate that our method exceeds in performance prior state-of-the-art monocular depth estimation and completion competitors. The source code will be available at https://github.com/ShuweiShao/NDDepth.Comment: Extension of previous work arXiv:2309.1059

Mapping Exoplanets

The varied surfaces and atmospheres of planets make them interesting places to live, explore, and study from afar. Unfortunately, the great distance to exoplanets makes it impossible to resolve their disk with current or near-term technology. It is still possible, however, to deduce spatial inhomogeneities in exoplanets provided that different regions are visible at different times---this can be due to rotation, orbital motion, and occultations by a star, planet, or moon. Astronomers have so far constructed maps of thermal emission and albedo for short period giant planets. These maps constrain atmospheric dynamics and cloud patterns in exotic atmospheres. In the future, exo-cartography could yield surface maps of terrestrial planets, hinting at the geophysical and geochemical processes that shape them.Comment: Updated chapter for Handbook of Exoplanets, eds. Deeg & Belmonte. 17 pages, including 6 figures and 4 pages of reference

Conductance of a STM contact on the surface of a thin film

The conductance of a contact, having a radius smaller than the Fermi wave length, on the surface of a thin metal film is investigated theoretically. It is shown that quantization of the electron energy spectrum in the film leads to a step-like dependence of differential conductance G(V) as a function of applied bias eV. The distance between neighboring steps in eV equals the energy level spacing due to size quantization. We demonstrate that a study of G(V) for both signs of the voltage maps the spectrum of energy levels above and below Fermi surface in scanning tunneling experiments.Comment: 15 pages, 5 figure