Geometric Quantities Research Articles

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

Hartle-Hawking state and its factorization in 3d gravity

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and “baby universe” operators in 3d gravity.

The mass gap in five dimensional Einstein–Gauss–Bonnet black holes: a geometrical explanation

It is well known that, unlike in higher dimensional general relativity (GR), we cannot have a black hole with an arbitrarily small mass in five dimensional Einstein–Gauss–Bonnet gravity. When we study the dynamical black hole formation via the radiation collapse in the radiating Boulware–Deser spacetime in five dimensions, the central zero mass singularity is weak, conical and naked, and the horizon forms only when a finite amount of matter, that depends on the coupling constant of the Gauss–Bonnet term, falls into the central singularity. To understand this phenomenon transparently and geometrically, we study the radiating Boulware–Deser spacetime in five dimensions using a 1+1+3 spacetime decomposition, for the first time. We find that the geometric and thermodynamic quantities can be expressed in terms of the gravitational mass and the Gauss–Bonnet (GB) parameter and separate each of them into their Gauss–Bonnet and matter parts. Drawing comparisons with five dimensional GR at every step, we explicitly show how the mass gap arises for a general mass function M(v) and what functions for M(v) make certain geometrical quantities well defined at the central singularity. We show in the case of self-similar radiation collapse in the modified theory, the central singularity is not a sink for timelike geodesics and is extendable. This clearly demonstrates how the GB invariant affects the nature of the final state of a continual collapse in this modified theory.

Two loop renormalization of scalar theories using a geometric approach

We derive a general formula for two-loop counterterms in Effective Field Theories (EFTs) using a geometric approach. This formula allows the two-loop results of our previous paper to be applied to a wide range of theories. The two-loop results hold for loop graphs in EFTs where the interaction vertices contain operators of arbitrarily high dimension, but at most two derivatives. We also extend our previous one-loop result to include operators with an arbitrary number of derivatives, as long as there is at most one derivative acting on each field. The final result for the two-loop counterterms is written in terms of geometric quantities such as the Riemann curvature tensor of the scalar manifold and its covariant derivatives. As applications of our results, we give the two-loop counterterms and renormalization group equations for the O(n) EFT to dimension six, the scalar sector of the Standard Model Effective Field Theory (SMEFT) to dimension six, and chiral perturbation theory to order p6.

Qualitative behaviour of higher-curvature gravity with boundary terms i.e the f(Q) gravity models by dynamical system analysis

The higher-curvature gravity with boundary terms i.e the f(Q) theories, grounded on non-metricity as a fundamental geometric quantity, exhibit remarkable efficacy in portraying late-time universe phenomena. The aim is to delineate constraints on two prevalent models within this framework, namely the Log-square-root model and the Hyperbolic tangent-power model, by employing the framework of Big Bang Nucleosynthesis (BBN). The approach involves elucidating deviations induced by higher-curvature gravity with boundary terms in the freeze-out temperature (Tf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{f}$$\\end{document}) concerning its departure from the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM evolution. Subsequently, constraints on pertinent model parameters are established by imposing limitations on |δTfTf|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert \\frac{\\delta T_{f}}{T_{f}}\\vert $$\\end{document} derived from observational bounds. This investigation employs dynamical system analysis, scrutinizing both background and perturbed equations. The study systematically explores the phase space of the models, identifying equilibrium points, evaluating their stability, and comprehending the system’s trajectory around each critical point. Moreover, the application of “Center Manifold Theory” facilitates the examination of stability properties at non-hyperbolic points. The principal findings of this analysis reveal the presence of a matter-dominated saddle point characterized by the appropriate matter perturbation growth rate. Subsequently, this phase transitions into a stable phase of a dark-energy-dominated, accelerating universe, marked by consistent matter perturbations. Overall, the study substantiates observational confrontations, affirming the potential of higher-curvature gravity with boundary terms as a promising alternative to the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM concordance model. The methodological approach underscores the significance of dynamical systems as an independent means to validate and comprehend the cosmological implications of these theories.

Giant Third-Order Nonlinear Hall Effect in Misfit Layer Compound (SnS)1.17(NbS2)3.

The nonlinear Hall effect (NLHE) holds immense significance in recognizing the band geometry and its potential applications in current rectification. Recent discoveries have expanded the study from second-order to third-order nonlinear Hall effect (THE), which is governed by an intrinsic band geometric quantity called the Berry Connection Polarizability tensor. Here we demonstrate a giant THE in a misfit layer compound, (SnS)1.17(NbS2)3. While the THE is prohibited in individual NbS2 and SnS due to the constraints imposed by the crystal symmetry and their band structures, a remarkable THE emerges when a superlattice is formed by introducing a monolayer of SnS. The angular-dependent THE and its scaling relationship indicate that the phenomenon could be correlated to the band geometry modulation, concurrently with the symmetry breaking. The resulting strength of THE is orders of magnitude higher compared to recent studies. Our work illuminates the modulation of structural and electronic geometries for novel quantum phenomena through interface engineering.

The geometry of inflationary observables: Lifts, flows, equivalence classes

The Eisenhart lift allows formulating the dynamics of a scalar field in a potential as pure geodesic motion in a curved field-space manifold involving an additional fictitious vector field. Making use of the formalism in the context of inflation, we show that the main inflationary observables can be expressed in terms of the geometrical properties of a two-dimensional uplifted field-space manifold spanned by the time derivatives of the scalar and the temporal component of the vector. This allows to abstract from specific potentials and models and describe inflation solely in terms of the flow of geometric quantities. Our findings are illustrated through several inflationary examples previously considered in the literature.

Geometric Aspects of Mixed Quantum States Inside the Bloch Sphere

When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.

A Discovery Tour in Random Riemannian Geometry

We study random perturbations of a Riemannian manifold (M,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extsf{M},\ extsf{g})$$\\end{document} by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields h∙:ω↦hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^\\bullet : \\omega \\mapsto h^\\omega $$\\end{document} will act on the manifold via the conformal transformation g↦gω:=e2hωg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{g}\\mapsto \ extsf{g}^\\omega := e^{2h^\\omega }\\,\ extsf{g}$$\\end{document}. Our focus will be on the regular case with Hurst parameter H>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>0$$\\end{document}, the critical case H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

A Simplified Method for Calculating Surface Area of Mammalian Erythrocytes.

Knowledge of the geometric quantities of the erythrocyte is useful in several physiological studies, both for zoologists and veterinarians. While the diameter and volume (MCV) are easily obtained from observations of blood smears and complete blood count, respectively, the thickness and surface area are instead much more difficult to measure. The precise description of the erythrocyte geometry is given by the equation of the oval of Cassini, but the formulas deriving from it are very complex, comprising elliptic integrals. In this article, three solids are proposed as models approximating the erythrocyte: sphere, cylinder and a spheroid with concave caps. The volumes and surface areas obtained with these models are compared to those effectively measured. The spheroid with concave caps gives the best approximation and can be used as a simple model to determine the erythrocyte surface area. With this model, a simple method that allows one to estimate the surface area by knowing only the diameter and MCV is proposed.

Possible Geometries for Precast Concrete Structures, through Discussing New Connections, Robotic Manufacturing and Re-Utilisation of the Concrete Elements

This study explores the potential use of new connections to shape precast building geometries, focusing on connection performance, robotic fabrication, and foldable structural elements. Three connection types, including coupled-bolts, hinges, and steel tubes, were initially proposed and assessed in beam and portal frame geometries. In contrast, the study introduces conceptual ideas; initial experimental and numerical studies were conducted to estimate connection capacities. Robotic fabrication for connecting elements to reused concrete and converting floor elements into beams was detailed, showcasing robotic technology’s performance and potential. These connections were employed in designing new precast element geometries, ranging from simple beams to multi-story buildings. Geometric properties and volume quantities of folded and opened geometries were studied using 37 CAD models. To properly discuss the joint performance reference, monolithic elements with exact dimensions were created for comparison. Despite varied connection capacity (38% to 100%), the steel tube exhibited the most desirable performance, resembling a monolithic element with an exact size. Some proposed foldable geometries showed a significant reduction (up to 7%) in element dimensions to facilitate transport and construction.

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive griding results in more accurate computation of geometric quantities such as x-points associated with the model.

The role of the areal parameters on turbulent flow over 2D Gaussian roughness

In the past decades, numerous efforts have been dedicated to establishing a direct correlation between a geometric parameter that represents wall roughness and the corresponding velocity reduction, known as the Roughness Function ΔU+. This reduction is influenced by various statistical measures of roughness height, including average roughness height, peak-to-valley roughness distance, roughness root mean square, and Effective Slope, among others. It has been demonstrated that a singular measure of roughness height cannot sufficiently predict the Roughness Function across all turbulent regimes. Consequently, many studies have concentrated on identifying a universal correlation between roughness geometry and the downward shift of the mean velocity profile. In this study, we investigated the correlation between various geometrical parameters and the Roughness Function using Large Eddy Simulation (LES) techniques in channel flows at a friction Reynolds number of Reτ=400. Given the complexity introduced by the random nature of irregular roughness, we explored specific aspects of the relationship between wall irregularities and the roughness function by studying 2D geometries. This approach allowed us to systematically investigate the impact of geometrical properties on the roughness function and isolate the effects of roughness density and coverage area. Several irregular rough surfaces, characterized by different average oscillations height, different distributions and different densities, were designed throughout 2D Gaussian functions. With the aim to find a universal correlation between the roughness geometry and the Roughness Function, new geometrical quantities were investigated, based on the global area occupied by the roughness A∗. The prediction of the roughness function can be thus obtained using apriori data. To predict the roughness function, we introduced a parameter called Effective Area (EA), which is derived from the correlation between the Effective Slope (ES) and the roughness area A∗. Our findings indicate that a single geometric parameter, whether ES or Area A, is insufficient to predict the roughness effect. Conversely, combining these two parameters enhances predictive accuracy, at least for the proposed roughness model. This improvement can be attributed to the ability of ES to interpret the roughness distribution and height, while the coverage area is effective in predicting roughness density over a flat plate.

On Crofton’s type formulas and the solid angle of convex sets

AbstractHere we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely:λ1(β,Ω)=minψ∈W1,p(Ω)∖{0}⁡∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN−1∫Ω|ψ|pdx, where p∈]1,+∞[, Ω is a bounded, anisotropic mean convex domain in RN, νΩ is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on RN. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ1 in the case β>0, and a upper bound in the case β<0. As a consequence, we prove, for β>0, a lower bound for λ1(β,Ω) in terms of the anisotropic inradius of Ω and, for β<0, an upper bound of λ1(β,Ω) in terms of β.

Sharp inequalities involving the Cheeger constant of planar convex sets

We are interested in finding sharp bounds for the Cheeger constant h via different geometrical quantities, namely the area |•|, the perimeter P, the inradius r, the circumradius R, the minimal width w and the diameter d. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets (P, h, r), (d, h, r) and (R, h, r) and describe some parts of the boundaries of the diagrams of the triplets (w, h, d), (w, h, R), (w, h, P), (w, h, |•|), (R, h, d) and (w, h, r).

On geometries and monodromies for branes of codimension two

We study geometries for the NS5-, the KK5- and the 522-branes of codimension two in type II and heterotic string theories. The geometries are classified by monodromies that each brane has. They are the B-, the general coordinate and the β-transformations of the spacetime metric, the B-field and the dilaton (and the gauge fields). We show that the monodromy nature appears also in the geometric quantities such as the curvature and the complex structures of spacetime. They are linearly realized in the doubled (generalized) structures in the doubled space.

Derivation and simulation of a two-phase fluid deformable surface model

To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator \square_{g} is known to be essentially self-adjoint. We define complex powers (\square_{g}-i\varepsilon)^{-\alpha} by functional calculus, and show that the trace density exists as a meromorphic function of \alpha . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine–Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.

A dynamical Thouless formula

In this paper we establish an abstract, dynamical Thouless-type formula for affine families of GL2(R) cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.