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Marko Sobak

Publications and source records attributed to Marko Sobak.

9 recordsLinked to original sources
arXiv2026

Static spherically symmetric electroweak models with fermions and gravity

In this article we provide a systematic reduction to static spherical symmetry for (classical) gauge theoretic models with structure group SU(2) x U(1), coupled to gravity. In particular, we derive the most general static spherically symmetric ansatz for the Einstein-Yang- Mills-Higgs-Dirac-Yukawa equation system. This model is reminiscent of the lepton sector of the Standard Model coupled to gravity. The resulting total system of ordinary differential equations is rather large, so that finding global solutions is non-trivial. For this reason, an open-source interactive solver GUI written in C++ is provided together with the article.

gr-qc↗
arXiv2026

The energy-momentum tensor of the Standard Model with applications to energy conditions

The Standard Model of elementary particle physics is one of the most successful models of contemporary physics, its predictions being in full agreement with experiments. In this manuscript we consider the Lagrangian of the Standard Model as a geometric variational problem on a globally hyperbolic manifold and derive the associated energy-momentum tensor in a geometric invariant way. As an application, we investigate the validity of various energy conditions that arise in general relativity.

math.DG↗
arXiv2026

Spherically symmetric Dirac-Yang-Mills pairs on Riemannian manifolds

In this paper we construct examples of spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with the structure group SU(2). This approach yields coupled solutions (i.e. the connection is not a Yang-Mills connection) and among them are solutions on S^1(r_1) x S^2(r_2) for certain radii r_1 and r_2. We further show how to use such pairs to induce Dirac-Yang-Mills pairs on Riemannian products of arbitrary dimension. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

math.DG↗
arXiv2025

Global critical points of the Standard Model on four-dimensional spacetimes of expanding type

The Standard Model of elementary particle physics is one of the most successful models of contemporary theoretical physics being in full agreement with experiments. However, its mathematical structure deserves further investigations both from a geometric and an analytic point of view. The aim of this manuscript is to provide a mathematically well-defined and self-contained description of the Standard Model in terms of gauge theory and differential geometry on globally hyperbolic manifolds. Within this setup we then prove the existence of a global solution for the Euler-Lagrange equations of the Standard Model (with the conformal Higgs potential) under the assumptions that the globally hyperbolic manifold is a four-dimensional spacetime of expanding type and small initial data. This is achieved by establishing a gauge-invariant energy estimate which is of independent mathematical interest.

math.DG↗
arXiv2023

Einstein-Yang-Mills wormholes haunted by a phantom field

In this article, we study wormhole spacetimes in the framework of the static spherically symmetric SU(2) Einstein-Yang-Mills theory coupled to a phantom scalar field. We show rigorously the existence of an infinite sequence of symmetric wormhole solutions, labelled by the number of zeros of the Yang-Mills potential. These solutions have previously been discovered numerically. Mathematically, the problem resembles the pure Einstein-Yang-Mills system for black hole initial conditions, which was well-studied in the 90s. The main difference in the present work is that the coupling to the phantom field adds a non-trivial degree of complexity to the analysis. After proving the existence of the symmetric wormhole solutions, we also present numerical evidence for the existence of asymmetric ones.

math-ph↗
arXiv2021

Explicit p-harmonic functions on rank-one Lie groups of Iwasawa type

We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.

math.DG↗
arXiv2020

$p$-Harmonic and Complex Isoparametric Functions on the Lie Groups $\mathbb{R}^m \ltimes \mathbb{R}^n$ and $\mathbb{R}^m \ltimes \mathrm{H}^{2n+1}$

In this paper we introduce the new notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper $p$-harmonic functions. We then apply this to construct the first known explicit proper $p$-harmonic functions on the Lie group semidirect products $\mathbb{R}^m \ltimes \mathbb{R}^n$ and $\mathbb{R}^m \ltimes \mathrm{H}^{2n+1}$, where $\mathrm{H}^{2n+1}$ denotes the classical $(2n+1)$-dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

math.DG↗
arXiv2019

Proper $r$-harmonic functions from Riemannian manifolds

We introduce a new method for constructing complex-valued $r$-harmonic functions on Riemannian manifolds. We then apply this method for the important semisimple Lie groups $SO(n)$, $SU(n)$, $Sp(n)$, $SL_n(R)$, $Sp(R,n)$, $SU(p,q)$, $SO(p,q)$, $Sp(p,q)$, $SO^*(2n)$ and $SU^*(2n)$.

math.DG↗