Scalar Laplacian Research Articles

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

A Selberg zeta function for warped AdS3 black holes

The Selberg zeta function and trace formula are powerful tools used to calculate kinetic operator spectra and quasinormal modes on hyperbolic quotient spacetimes. In this article, we extend this formalism to non-hyperbolic quotients by constructing a Selberg zeta function for warped AdS3 black holes. We also consider the so-called self-dual solutions, which are of interest in connection to near-horizon extremal Kerr. We establish a map between the zeta function zeroes and the quasinormal modes on warped AdS3 black hole backgrounds. In the process, we use a method involving conformal coordinates and the symmetry structure of the scalar Laplacian to construct a warped version of the hyperbolic half-space metric, which to our knowledge is new and may have interesting applications of its own, which we describe. We end by discussing several future directions for this work, such as computing 1-loop determinants (which govern quantum corrections) on the quotient spacetimes we consider, as well as adapting the formalism presented here to more generic orbifolds.

Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.

Laplacian on fuzzy de Sitter space

We study details of geometry of noncommutative de Sitter space: we determine the Riemann and Ricci curvature tensors, the energy and the Laplacian. We find, in particular, that fuzzy de Sitter space is an Einstein space, R ab = −3ζη ab . The Laplacian, defined in the noncommutative frame formalism, is not Hermitian and gives nonunitary evolution. When symmetrically ordered, it has the usual quadratic form Δ = Π a Π a (when acting on functions in representation space, ): we find its eigenstates and discuss its spectrum. This result is a first step in a study of the scalar field Laplacian, Δ = [Π a , [Π a , ]], and its propagator.

Relative scale separation in orbifolds of S2 and S5

In orbifold vacua containing an Sq/Γ factor, we compute the relative order of scale separation, r, defined as the ratio of the eigenvalue of the lowest-lying Γ-invariant state of the scalar Laplacian on Sq, to the eigenvalue of the lowest-lying state. For q = 2 and Γ finite subgroup of SO(3), or q = 5 and Γ finite subgroup of SU(3), the maximal relative order of scale separation that can be achieved is r = 21 or r = 12, respectively. For smooth S5 orbifolds, the maximal relative scale separation is r = 4.2. Methods from invariant theory are very efficient in constructing Γ-invariant spherical harmonics, and can be readily generalized to other orbifolds.

Scalar Pre-potentials for Spinor and Tensor Fields on Spacetime

We review a technique for solving a class of classical linear partial diferential systems of relevance to physics in Minkowski spacetime. All the equations are amenable to analysis in terms of complex solutions in the kernel of the scalar Laplacian and a complexifed Hertz potential. The complexifcation prescription ensures the existence of regular physical solutions with chirality and propagating, non-singular, pulse-like characteristics that are bounded in all three spatial dimensions. The technique is applied to the source-free Maxwell, Bopp-Landé-Podolsky and linearised Einstein feld systems, and particular solutions are used for constructing classical models describing single-cycle laser pulses and a mechanism is discussed for initiating astrophysical jets. Our article concludes with a brief introduction to spacetime Cliford algebra ideals that we use to represent spinor felds. We employ these to demonstrate how the same technique used for tensor felds enables one to construct new propagating, chiral, non-singular, pulse-like spinor solutions to the massless Dirac equation in Minkowski spacetime.

New eigenvalue estimates involving Bessel functions

Given a compact Riemannian manifold (Mn, g) with boundary ∂M, we give an estimate for the quotient R ∂M f dµg R M f dµg , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Indepen[1]dently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

On gradient estimates for heat kernels

We study pointwise and Lp gradient estimates of the heat kernels of both the scalar Laplacian, as well as the Hodge Laplacian on k-forms, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. Such heat kernel estimates have already been obtained by the author, together with Coulhon and Sikora, provided certain L 2-cohomology spaces are trivial. This is however a strong topological assumption, and it is desirable to weaken it. The main point of the current work is to investigate what happens when these L 2-cohomology spaces are non-trivial. We find that the answer depends on some Lq integrability properties of L 2-harmonic forms.

High frequency limits for invariant Ruelle densities

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant state converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

AbstractA parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

A Simple Discretization of the Vector Dirichlet Energy

AbstractWe present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.

Unitarization from geometry

We study the perturbative unitarity of scattering amplitudes in general dimensional reductions of Yang-Mills theories and general relativity on closed internal manifolds. For the tree amplitudes of the dimensionally reduced theory to have the expected high-energy behavior of the higher-dimensional theory, the masses and cubic couplings of the Kaluza-Klein states must satisfy certain sum rules that ensure there are nontrivial cancellations between Feynman diagrams. These sum rules give constraints on the spectra and triple overlap integrals of eigenfunctions of Laplacian operators on the internal manifold and can be proven directly using Hodge and eigenfunction decompositions. One consequence of these constraints is that there is an upper bound on the ratio of consecutive eigenvalues of the scalar Laplacian on closed Ricci-flat manifolds with special holonomy. This gives a sharp bound on the allowed gaps between Kaluza-Klein excitations of the graviton that also applies to Calabi-Yau compactifications of string theory.

Separation Problem for Bi-Harmonic Differential Operators in Lp\u2212 spaces on Manifolds

Consider the bi-harmonic differential expression of the form A=triangle _{M}^{2}+q on a manifold of bounded geometry (M,g) with metric g, where △M is the scalar Laplacian on M and q≥0 is a locally integrable function on M.In the terminology of Everitt and Giertz, the differential expression A is said to be separated in Lp(M), if for all u∈Lp(M) such that Au∈Lp(M), we have qu∈Lp(M). In this paper, we give sufficient conditions for A to be separated in Lp(M),where 1<p<∞.

A scaled boundary finite element formulation for poroelasticity

SummaryThis paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.

Mathcal{N} = 4 superconformal bootstrap of the K3 CFT

We study two-dimensional (4, 4) superconformal field theories of central charge c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS mathcal{N} = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks with c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and find numerically an upper bound on this gap that is saturated by the A1 mathcal{N} = 4 cigar CFT. We also derive an analytic upper bound on the first nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS mathcal{N} = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of sufficiently low scaling dimension in three and four dimensional CFTs.

Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class

AbstractWe develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

Exact Poincaré constants in two‐dimensional annuli

We provide precise estimates of the Poincaré constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d‐annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non‐dimensional setting each annulus is defined via two concentrical circles with radii and . Additionally, corresponding problems on domains , the 2d‐annuli from , are investigated ‐ for comparison but also to provide limits for . In particular, the Green's function of the Laplacian on with vanishing Dirichlet traces on is used to show that for the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so‐called small‐gap limit for .

Nonlinear nonhomogeneous periodic problems

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Caratheodory reaction. We show that it has at least three solutions, two of constant sign and the third nodal. In the particular case of the scalar \({p-}\)Laplacian and with a parametric reaction of equidiffusive type, we show that three solutions with precise sign exist if the parameter \({\lambda > \widehat{\lambda}_1(p)=}\) the first nonzero eigenvalue of the periodic scalar Laplacian. Finally, in the semilinear case \({(p=2),}\) we show that there is a second nodal solution, for a total of four nontrivial solutions all with sign information.

Bounding the heat trace of a Calabi-Yau manifold

The SCHOK bound states that the number of marginal deformations of certain two-dimensional conformal field theories is bounded linearly from above by the number of relevant operators. In conformal field theories defined via sigma models into Calabi-Yau manifolds, relevant operators can be estimated, in the point-particle approximation, by the low-lying spectrum of the scalar Laplacian on the manifold. In the strict large volume limit, the standard asymptotic expansion of Weyl and Minakshisundaram-Pleijel diverges with the higher-order curvature invariants. We propose that it would be sufficient to find an a priori uniform bound on the trace of the heat kernel for large but finite volume. As a first step in this direction, we then study the heat trace asymptotics, as well as the actual spectrum of the scalar Laplacian, in the vicinity of a conifold singularity. The eigenfunctions can be written in terms of confluent Heun functions, the analysis of which gives evidence that regions of large curvature will not prevent the existence of a bound of this type. This is also in line with general mathematical expectations about spectral continuity for manifolds with conical singularities. A sharper version of our results could, in combination with the SCHOK bound, provide a basis for a global restriction on the dimension of the moduli space of Calabi-Yau manifolds.

Spectral theory for the Weil–Petersson Laplacian on the Riemann moduli space

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric g_{\mathrm{WP}} on \mathcal M_\gamma , the Riemann moduli space of surfaces of genus \gamma &gt; 1 . This space has a singular compactification with respect to g_{\mathrm{WP}} , and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function for this spectrum.