arXiv subjects

Eduardo Hafemann

Publications and source records attributed to Eduardo Hafemann.

5 recordsLinked to original sources

arXiv2026

Coordinate isoperimetric mass and a positive mass theorem for distributional scalar curvature

Extending the well-known Riemannian positive mass theorem to low-regularity metrics is an ongoing effort, starting from the early 2000s, that has recently seen renewed interest. Contrary to many other approaches we do not wish to assume smoothness at infinity nor to introduce a new definition of mass tailored to our approach, but rather go back to the coordinate isoperimetric mass originally proposed by Huisken. We establish new asymptotic expressions for the coordinate isoperimetric mass of continuous asymptotically flat metrics, which constitute the main ingredient in proving a finiteness criterion, coordinate invariance and equality to the generalized ADM mass of Lee and LeFloch under reasonable assumptions. In particular, for $W^{1,p}_{-τ}$ asymptotically flat metrics, with $p>n$, $τ>(n-2)/2$, and nonnegative distributional scalar curvature, the coordinate isoperimetric mass is indeed coordinate invariant, and we establish a positive mass theorem in this regularity without any topological assumptions aside from asymptotic flatness. Our proof relies on a density theorem similar to that of Lee, Lesourd, and Unger, but under a weaker regularity and decay assumption, which may be of independent interest.

math.DG↗

arXiv2026

A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity in the metric-space sense is obtained via the volume comparison theory of $\sf{RCD}$-spaces after establishing that zero ADM mass implies Ricci flatness, which fundamentally relies on these improved estimates. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set.

math.DG↗

arXiv2026

A piecewise constant levelset approach for semi-blind deconvolution: Application to barcode decoding

We consider a semi-blind deconvolution problem arising in the decoding of blurred linear barcodes. Building on the Piecewise Constant Level Set (PCLS) framework introduced in [De\,Cezaro et al., Inv.\,Probl., 29 (2013), 015003], we propose and analyze a solution method based on augmented Lagrangians to obtain stable approximate solutions to the corresponding inverse problem with respect to noisy measurements. We establish the existence of generalized multipliers for the augmented Lagrangian functional under consideration, as well as the absence of duality gaps. These results provide the theoretical foundation required to prove regularization properties of the approximate solutions produced by the proposed strategy. Furthermore, we present an associated ADMM-type iterative scheme for the explicit computation of approximate barcodes. Numerical experiments are carried out for various variance values (responsible for the blurred effect) and several levels of noise, validating the effectiveness of the proposed method.

math.NA↗

arXiv2025

Hawking's singularity theorem for Lipschitz Lorentzian metrics

We prove Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity. The proof rests on (1) new estimates for the Ricci curvature of regularising smooth metrics that are based upon a quite general Friedrichs-type lemma and (2) the replacement of the usual focusing techniques for timelike geodesics -- which in the absence of a classical ODE-theory for the initial value problem are no longer available -- by a worldvolume estimate based on a segment-type inequality that allows one to control the volume of the set of points in a spacelike surface that possess long maximisers.

math.DG↗

arXiv2025

Penrose inequality for integral energy conditions

The classical Penrose inequality, a relation between the ADM mass and the area of any cross section of the black hole event horizon, was introduced as a test of the weak cosmic censorship conjecture: if it fails, the trapped surface is not necessarily behind the event horizon and a naked singularity could form. Since that original derivation, a variety of proofs have developed, mainly focused on the initial data formulation on maximal spacelike slices of spacetime. Most of these proofs are applicable only for classical fields, as the energy conditions required are violated in the context of quantum field theory. In this work we provide two generalizations of the Penrose inequality for spherically symmetric spacetimes: a proof of a classical Penrose inequality using initial data and an average energy condition, and a proof of a modified Penrose inequality for evaporating black holes with a connection to the weak cosmic censorship conjecture. The latter case could also be applicable to quantum fields as it uses a condition inspired by quantum energy inequalities. Finally, we provide physically motivated examples for both.

gr-qc↗