Isometry groups of Lorentzian manifolds of finite volume and The local geometry of compact homogeneous Lorentz spaces

Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, we also describe the geometric structure of the manifolds if they are compact. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation, curvatures and holonomy. Especially, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.Comment: 130 page

The convergence of discrete period matrices

We study compact polyhedral surfaces as Riemann surfaces and their discrete counterparts obtained through quadrilateral cellular decompositions and a linear discretization of the Cauchy-Riemann equation. By ensuring uniformly bounded interior and intersection angles of diagonals, we establish the convergence of discrete Dirichlet energies of discrete harmonic differentials with equal black and white periods to the Dirichlet energy of the corresponding continuous harmonic differential with the same periods. This convergence also extends to the discrete period matrix, with a description of the blocks of the complete discrete period matrix in the limit. Moreover, when the quadrilaterals have orthogonal diagonals, we observe convergence of discrete Abelian integrals of the first kind. Adapting the quadrangulations around conical singularities allows us to improve the convergence rate to a linear function of the maximum edge length.Comment: 33 pages, 3 figure

Balancing climate goals and biodiversity protection: legal implications of the 30x30 target for land-based carbon removal

This article examines the legal conflicts between land-based carbon dioxide removal (CDR) strategies and the establishment of protected areas through the lens of international environmental law. We argue that the 2022 Global Biodiversity Framework's “30x30” target—which aims to protect 30% of global terrestrial and marine areas by 2030—constitutes a “subsequent agreement” under international law and thus clarifies the legal scope and content of the obligation to establish protected areas under Article 8 of the Convention on Biological Diversity (CBD). Since states have pledged 120 million square kilometers for land-based CDR, these commitments potentially conflict with the “30x30” target, especially if global cropland for food production is to be maintained. Consequently, some land-based CDR strategies may directly or indirectly impede the achievement of the “30x30” target, which could be deemed inconsistent with international law. However, as all international environmental law operates in a continuum, this does not imply that land-based CDR should be categorically ruled out. Rather, states should focus on emission reductions and implementing CDR options that provide the most co-benefits to climate mitigation and biodiversity protection efforts

Winkeltreue zahlt sich aus

Nicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen nur mit einer überschaubaren Datenmenge zu beschreiben. Entsprechende Theorien werden in der diskreten Mathematik untersucht. Im Folgenden lade ich Sie auf eine Reise in die faszinierende Welt der winkeltreuen Abbildungen ein