A Positive Mass Theorem for Manifolds with Boundary

We derive a positive mass theorem for asymptotically flat manifolds with boundary whose mean curvature satisfies a sharp estimate involving the conformal Green's function. The theorem also holds if the conformal Green's function is replaced by the standard Green's function for the Laplacian operator. As an application, we obtain an inequality relating the mass and harmonic functions that generalizes H. Bray's mass-capacity inequality in his proof of the Riemannian Penrose conjecture.Comment: 15 pages, to appear in Pacific J. Mat

Stability of Llarull's theorem in all dimensions

We prove stability of Llarull's theorem in all dimensions using spin geometry. Our results are stated in terms of both intrinsic flat convergence and $C^0$ convergence outside a small set.Comment: Comments welcom

The case of equality for the spacetime positive mass theorem

The rigidity of the spacetime positive mass theorem states that an initial data set $(M,g,k)$ satisfying the dominant energy condition with vanishing mass can be isometrically embedded into Minkowski space. This has been established by Beig-Chru\'sciel and Huang-Lee under additional decay assumptions for the energy and momentum densities $\mu$ and $J$. In this note we give a new and elementary proof in dimension 3 which removes these additional decay assumptions. Our argument uses spacetime harmonic functions and Liouville's theorem. We also provide an alternative proof based on the Killing development of $(M,g,k)$.Comment: Comments welcome

Holographic topometry for a dense visualization of soft tissue for facial reconstruction

Facial reconstruction can be used as a forensic technique to identify a person, when no other identification method is applicable. The facial soft tissue thickness applied to the skull is crucial when performing an accurate facial reconstruction. Historically, scientists developed several techniques to measure the soft tissue of the face. It was their aim, to build a database of a unique point-set, differentiated by gender, age, ethnic origin, BMI. All used a limited number of landmarks and an inaccurate measuring technique. We developed a contact-free and precise measuring technique, using low-dose CT and holographic data. Due to the extremely short exposure time, the holographic measurement is very precise. We lay out our first experiences to create a facial soft tissue layer map of the fac

A generalization of Geroch's conjecture

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called $m$-intermediate curvature), and use stable weighted slicings to show that for $n \leq 7$ the manifolds $N^n = M^{n-m} \times \mathbb{T}^m$ do not admit a metric of positive $m$-intermediate curvature.Comment: final version; to appear in Comm. Pure. Appl. Mat

causalgraph: A Python Package for Modeling, Persisting and Visualizing Causal Graphs Embedded in Knowledge Graphs

This paper describes a novel Python package, named causalgraph, for modeling and saving causal graphs embedded in knowledge graphs. The package has been designed to provide an interface between causal disciplines such as causal discovery and causal inference. With this package, users can create and save causal graphs and export the generated graphs for use in other graph-based packages. The main advantage of the proposed package is its ability to facilitate the linking of additional information and metadata to causal structures. In addition, the package offers a variety of functions for graph modeling and plotting, such as editing, adding, and deleting nodes and edges. It is also compatible with widely used graph data science libraries such as NetworkX and Tigramite and incorporates a specially developed causalgraph ontology in the background. This paper provides an overview of the package's main features, functionality, and usage examples, enabling the reader to use the package effectively in practice