Bathymetry of Lake Constance - State-of-the-art in surveying a large lake

In 2014 the by far largest German lake has been newly surveyed. The transnational project is funded by the European Union and delivers a detailed 3D-model of the lake- floor. The German project name is »Tiefenschärfe – Hochauflösende Vermessung Bo- densee«, which in English roughly means: high-resolution survey of Lake Constance. The German term »Tiefenschärfe« (in optics and photography: depth of field) plays with the meanings of »Tiefe« (depth) and »Schärfe« (sharpness). The result of the sur- vey shall be a clear and sharp image of the deep and shallow lake- floor. At present the LiDAR and multibeam data are still processed, but first results are presented in this article.

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.