Curvature estimates for stable marginally trapped surfaces

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical insterest, such as dynamical horizons.Comment: 28 pages. This is a major rework of the previous version. It extends the curvature estimates to no longer require global area bounds. In addition some mistakes were correcte

Large isoperimetric surfaces in initial data sets

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.Comment: 29 pages. All comments welcome! This is the final version to appear in J. Differential Geo

Jenkins-Serrin type results for the Jang equation

Let (M, g, k) be an initial data set for the Einstein equations of general relativity. We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an ingredient in the proof of the spacetime positive mass theorem given by L.-H. Huang, D. Lee, R. Schoen and the first author. We show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end, provided that this complement contains no weakly past outer trapped regions. The graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non-trivial way. We prove the existence of a Scherk-type solution of the Jang equation outside the union of all weakly future or past outer trapped regions in the initial data set. This result is a natural exterior analogue for the Jang equation of the classical Jenkins--Serrin theory. We extend and complement existence theorems by Jenkins-Serrin, Spruck, Nelli-Rosenberg, Hauswirth-Rosenberg-Spruck, and Pinheiro for Scherk-type constant mean curvature graphs over polygonal domains in (M, g), where (M, g) is a complete Riemannian surface. We can dispense with the a priori assumptions that a sub solution exists and that (M, g) has particular symmetries. Also, our method generalizes to higher dimensions.Comment: All comments welcome. This is the final version to appear in J. Differential Geo

The time evolution of marginally trapped surfaces

In previous work we have shown the existence of a dynamical horizon or marginally trapped tube (MOTT) containing a given strictly stable marginally outer trapped surface (MOTS). In this paper we show some results on the global behavior of MOTTs assuming the null energy condition. In particular we show that MOTSs persist in the sense that every Cauchy surface in the future of a given Cauchy surface containing a MOTS also must contain a MOTS. We describe a situation where the evolving outermost MOTS must jump during the coalescence of two seperate MOTSs. We furthermore characterize the behavior of MOTSs in the case that the principal eigenvalue vanishes under a genericity assumption. This leads to a regularity result for the tube of outermost MOTSs under the genericity assumption. This tube is then smooth up to finitely many jump times. Finally we discuss the relation of MOTSs to singularities of a space-time.Comment: 21 pages. This revision corrects some typos and contains more detailed proofs than the original versio

An AI Chatbot for Explaining Deep Reinforcement Learning Decisions of Service-oriented Systems

Deep Reinforcement Learning (Deep RL) is increasingly used to cope with the open-world assumption in service-oriented systems. Deep RL was successfully applied to problems such as dynamic service composition, job scheduling, and offloading, as well as service adaptation. While Deep RL offers many benefits, understanding the decision-making of Deep RL is challenging because its learned decision-making policy essentially appears as a black box. Yet, understanding the decision-making of Deep RL is key to help service developers perform debugging, support service providers to comply with relevant legal frameworks, and facilitate service users to build trust. We introduce Chat4XAI to facilitate the understanding of the decision-making of Deep RL by providing natural-language explanations. Compared with visual explanations, the reported benefits of natural-language explanations include better understandability for non-technical users, increased user acceptance and trust, as well as more efficient explanations. Chat4XAI leverages modern AI chatbot technology and dedicated prompt engineering. Compared to earlier work on natural-language explanations using classical software-based dialogue systems, using an AI chatbot eliminates the need for eliciting and defining potential questions and answers up-front. We prototypically realize Chat4XAI using OpenAI's ChatGPT API and evaluate the fidelity and stability of its explanations using an adaptive service exemplar.Comment: To be published at 21st Int'l Conference on Service-Oriented Computing (ICSOC 2023), Rome, Italy, November 28-December 1, 2023, ser. LNCS, F. Monti, S. Rinderle-Ma, A. Ruiz Cortes, Z. Zheng, M. Mecella, Eds., Springer, 202