arXiv research

Defines and computes a generalized spectral action for Lorentz warped products.

Extends Einstein-Hilbert action to higher-order spectral triples.

Extends Einstein-Hilbert functional definition for stable manifolds.

The paper calculates variations of Einstein-Hilbert action on CR manifolds.

The paper studies Einstein-Hilbert action on complex manifolds.

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…

Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.

In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on $n$-dimensional, $n\geq 3$, asymp…

New formula for Lichnerowicz Laplacian on homogeneous spaces.

The paper studies Einstein-Hilbert functional and its relation to K-semistability.

Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.

Paper proves convergence of MDL to Einstein-Hilbert with boundary term.

A new framework for Einstein-Hilbert action with topological variations.

Study of Einstein-Hilbert action on metric-affine spaces with connections.

A first-order Lagrangian $L^\nabla$ variationally equivalent to the second-order Einstein-Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by $L^\nabla$ is proved to be regular and its H…

New stable metric found on a complex space.

We discuss Levi-Civita connections on Courant algebroids. We define an appropriate generalization of the curvature tensor and compute the corresponding scalar curvatures in the exact and heterotic case, leading to generalized (bosonic) Einstein-Hilbert type of actions known from supergravity. In particular, we carefull…

The study examines stability of specific geometric flows.

In this paper, we studied the full Einstein-Hilbert actions with respect to non-symmetric metrics and the corresponding torsion. The first concrete result in this paper are the general formulae for pressure and density with respect to the Madsen's article (the equation (3.1), in [10]). Based on these results, we obtain…

We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient f…

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a dis…

Paper studies stability of generalized Ricci solitons with mathematical rigor.

We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki mani…

By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the…

Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Stud…

Develops Palatini formalism for pseudo-Finsler metrics, recovering classical results.

Study on stability of Einstein metrics on symmetric spaces.

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied e…

Study stability of Einstein manifolds with boundary.

Using the results and techniques of a previous paper where we proved the quantization of gravity we extend the former result by adding a Yang-Mills functional and a Higgs term to the Einstein-Hilbert action.

Certain curvature conditions for stability of Einstein manifolds with respect to the Einstein-Hilbert action are given. These conditions are given in terms of quantities involving the Weyl tensor and the Bochner tensor. In dimension six, a stability criterion involving the Euler characteristic is given.

Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on $TM \oplus Λ^{p} T^{\ast}M$ is an example leading to a scalar curvature of the form $R + H^2$ for a closed $(p+2)$-form $H$.

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…

HR in 8D encodes unique conformal gravity with negative curvature.

In this article we introduce $A$-valued Einstein-Hilbert-Palatini functional ($A$-EHP) over a n-manifold $M$, where $A$ is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if $A$ is weak $(k,s)$-solvable, then $A$-EHP is non-…

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and St…

The paper extends Einstein condition to 4-manifolds using Hodge splittings.

The Yamabe invariant is linked to static potentials and eigenvalues.

Proves stability of Einstein metrics on specific symmetric spaces.

This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, den…

In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}.…

This article considers the quasi-local energy in reference to a general static spacetime. We follow the approach developed by the authors in [19, 20, 7, 9] and define the quasi-local energy as a difference of surface Hamiltonians, which are derived from the Einstein-Hilbert action. The new quasi-local energy provides a…

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consis…

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski m…

The paper studies Einstein metrics on specific manifolds and their rigidity properties.

Surveying stability and deformation of Einstein metrics.

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local indepe…

We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures th…