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In this work, we have obtained exact solutions of Einstein equations for static and axially symmetric magnetized matter, specifically in plane-symmetric and almost-plane symmetric cases. Although these solutions impose constraints on the…
We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are K\"{a}hler. The type II metrics are ALH whose total energy can be negative.
Bianchi-IX four metrics are $SU(2)$ invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler Top, while the curvature-wise self-dual case yields the Ricci flat classical…
Continuous unitary transformations are a powerful tool to extract valuable information out of quantum many-body Hamiltonians, in which the so-called flow equation transforms the Hamiltonian to a diagonal or block-diagonal form in second…
We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the…
In this note, we review various seemingly different ways of obtaining Eguchi-Hanson metric with or without a cosmological constant term. Interestingly, the conformal class of metric corresponding to hyperbolic $n$-monopole solution obtained…
Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description…
We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics…
Examination of the Einstein energy-momentum relationship suggests that simple unbound forms of matter exist in a four-dimensional Euclidean space. Position, momentum, velocity, and other vector quantities can be expressed as Euclidean…
We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components. In…
In this article, we construct explicit analytical exact solutions to the six and higher dimensional Einstein-Maxwell theory. In all solutions, a subspace of the metric is the Eguchi-Hanson space where the metric functions are completely…
In this paper, we propose a new MUSCL scheme by combining the ideas of the Kurganov and Tadmor scheme and the so-called Deviation method which results in a well-balanced finite volume method for the hyperbolic balance laws, by evolving the…
This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact…
This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…
We find all orthogonal metrics where the geodesic Hamilton-Jacobi equation separates and the Riemann curvature tensor satisfies a certain equation (called the diagonal curvature condition). All orthogonal metrics of constant curvature…
Locally rotationally symmetric perfect fluid solutions of Einstein's gravitational equations are matched along the hypersurface of vanishing pressure with the NUT metric. These rigidly rotating fluids are interpreted as sources for the…
In this paper, we are concerned with the energy and magnetic helicity conservation of weak solutions for both the electron and Hall magnetohydrodynamic equations. Various sufficient criteria to ensure the energy and magnetic helicity…
We address a class of schemes for the Euler equations with the following features: the space discretization is staggered, possible upwinding is performed with respect to the material velocity only and the internal energy balance is solved,…
In this paper, we study the \texttt{Ehlers' transformation} (sometimes called gravitational duality rotation) for \texttt{reciprocal} static metrics. First we introduce the concept of reciprocal metric. We prove a theorem which shows how we…
The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution…