Bochner Laplacian Research Articles

Bochner Laplacian and Bergman kernel expansion of semipositive line bundles on a Riemann surface

We generalize the results of Montgomery (Commun Math Phys 168:651–675, 1995) for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion for semipositive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

Coarse geometric kernels for networks embedding

We develop embedding kernels based on the Forman–Ricci curvature and intertwined Bochner–Laplacian and employ them for the detection of the coarse structure of networks, as well as for network visualization with applications to support-vector machines (SVMs).

Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle

Review on Spectral asymptotics for the semiclassical Bochner Laplacian of a line bundle

Berezin–Toeplitz Quantization on Symplectic Manifolds of Bounded Geometry

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.

Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

In this paper, we construct a family of Berezin–Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers L^p of the prequantum line bundle L asymptotically splits into clusters of size {\mathcal O}(p^{3/4}) around the points p\Lambda , where \Lambda is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigenvalues from the cluster.We show that it provides a Berezin–Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.

The Spectral Density Function of the Renormalized Bochner Laplacian on a Symplectic Manifold

We consider the renormalized Bochner Laplacian acting on tensor powers of a positive line bundle on a compact symplectic manifold. We derive an explicit local formula for the spectral density function in terms of coefficients of the Riemannian metric and symplectic form.

On the Bochner Laplacian operator on theta line bundle over quasi-tori

In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; Then, we give a concrete description of each eigenspace in terms of Hermite and complex Hermite polynomials. In particular, an explicit description of the L2-holomorphic sections on the above line bundle is presented as the eigenspace of the magnetic Laplacian corresponding to the least eigenvalue. Also by using the periodization principle, the associated invariant integral operators are discussed.

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, different from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier–Stokes–Fourier equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

Optimal convergence speed of Bergman metrics on symplectic manifolds

It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. This result implies the generalization to the almost-K\"ahler case of the lower bounds on the Calabi functional given by Donaldson for K\"ahler manifolds, as shown by Lejmi and Keller.

On asymptotic expansions of generalized Bergman kernels on symplectic manifolds

A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed.

Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds

We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic manifold. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product.

Evolution of the Bochner Laplacian for magnetograms of solar active regions

A search for powerful flare predictors remains a problem of current interest in solar—terrestrial physics. The magnetograms (LOS SDO/HMI instrument) for active regions (ARs) 1158, 1166, 1283, and 1520, which produced an X-ray flare located near the central meridian, are analyzed. АR 1654, which was rather complex but generated only M flares near western limb, remaining quiet during the passage over the disk, has been analyzed in a similar manner for comparison. The combinatorial Bochner Laplacian was used as a complexity descriptor. We calculate it for each magnetogram, which was converted into the magnetic energy density of the longitudinal field component, and analyze its maximal spatial variation. It has been shown that the maximal spatial Laplacian values trace the neutral line during AR evolution, demonstrating sharp peaks before and after the flare in this case. Although this signature has no established statistical reliability, it can be interesting as an effective parameter when flares are predicted.

Sobolev Spaces and Bochner Laplacian on Complex Projective Varieties and Stratified Pseudomanifolds

Let $$V\subset {\mathbb {C}}{\mathbb {P}}^n$$ be an irreducible complex projective variety of complex dimension v and let g be the Kähler metric on $${{\mathrm{reg}}}(V)$$ , the regular part of V, induced by the Fubini–Study metric of $${\mathbb {C}}{\mathbb {P}}^n$$ . In (J Am Math Soc 8:857–877, 1995) Li and Tian proved that $$W^{1,2}_0({{\mathrm{reg}}}(V),g)=W^{1,2}({{\mathrm{reg}}}(V),g)$$ , that the natural inclusion $$W^{1,2}({{\mathrm{reg}}}(V),g)\hookrightarrow L^2({{\mathrm{reg}}}(V),g)$$ is a compact operator and that the heat operator associated with the Friedrich extension of the scalar Laplacian $$\Delta _0:C^{\infty }_c({{\mathrm{reg}}}(V))\rightarrow C^{\infty }_c({{\mathrm{reg}}}(V))$$ , that is, $$e^{-t\Delta _0^{{\mathcal {F}}}}:L^2({{\mathrm{reg}}}(V),g)\rightarrow L^2({{\mathrm{reg}}}(V),g)$$ , is a trace class operator. The goal of this paper is to provide an extension of the above result to the case of Sobolev spaces of sections and symmetric Schrödinger type operators with potential bounded from below where the underlying Riemannian manifold is the regular part of a complex projective variety endowed with the Fubini–Study metric or the regular part of a stratified pseudomanifold endowed with an iterated edge metric.

Generalized Laplacian for magnetograms of solar active region as possible predictor of strong flare.

Search for predictors of strong flare produced in solar active region (AR) is important application of solar physics. Here we consider the sequence of magnetogram (LOS SDO/HMI instrument) for AR 2034, 2035 and 2036 (April 2014). All three AR were observed on the Sun at about the same time, characterized by low probability of flare events according to official forecasts of NOAA, but 2036 still produced X1-flare near the center of solar disc (April 18). We propose that Generalized Laplacian is a descriptor which could help predict this and similar events. The Laplacian is associated with the flow of Ricci curvature and with topological invariants of the observed field - Betti numbers for compact manifolds. Using discrete version of Morse theory, we consider each pixel of energy flux (B2) image as a simplex and calculate its combinatorial Bochner Laplacian. It was found that maximum of Laplacian is located near AR polarity inversion line. Evolution of total spatial variation of the Laplacian has a number of maxima in time for each of examined AR. However, the maxima in AR 2035 and AR 2034 have relatively low amplitude, while the highest maximum prefaced X1 flare in AR 2036 by about 29 hours.

Equivariant heat invariants of the laplacian and nonmininmal operators on differential forms

In this paper, we compute the first two equivariant heat kernel coefficients of the Bochner Laplacian on differential forms. The first two equivariant heat kernel coefficients of the Bochner Laplacian with torsion are also given. We also study the equivariant heat kernel coefficients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.

Noncommutative Bloch analysis of Bochner Laplacians with nonvanishing gauge fields

Given an invariant gauge potential and a periodic scalar potential V ̃ on a Riemannian manifold M ̃ with a discrete symmetry group Γ , consider a Γ -periodic quantum Hamiltonian H ̃ = − Δ ̃ B + V ̃ where Δ ̃ B is the Bochner Laplacian. Both the gauge group and the symmetry group Γ can be noncommutative, and the gauge field need not vanish. On the other hand, Γ is supposed to be of type I. With any unitary representation Λ of Γ one associates a Hamiltonian H Λ = − Δ B Λ + V on M = M ̃ / Γ where V is the projection of V ̃ onto M . We describe a construction of the Bloch decomposition of H ̃ into a direct integral whose components are H Λ , with Λ running over the dual space Γ ˆ . The evolution operator and the resolvent decompose correspondingly. Conversely, given Λ ∈ Γ ˆ , one can express the propagator K t Λ ( y 1 , y 2 ) (the kernel of exp ( − i t H Λ ) ) in terms of the propagator K ̃ t ( y 1 , y 2 ) (the kernel of exp ( − i t H ̃ ) ) as a weighted sum over Γ . Such a formula is known in theoretical physics for the case when the gauge field vanishes and M ̃ is a universal covering space of a multiply connected manifold M . We show that these constructions are mutually inverse. Analogous formulas exist for resolvents and their kernels (Green functions) as well.

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression $${H=\nabla^*\nabla+V}$$ , where $${\nabla}$$ is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L p (E), where 1 < p < +∞. We study the same problem for the operator Δ M + V in L p (M), where 1 < p < ∞, Δ M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.

Eigenvalue estimates for the Bochner Laplacian and harmonic forms on complete manifolds

We study the set of eigenvalues of the Bochner Laplacian on a geodesic ball of an open manifold M, and find lower estimates for these eigenvalues when M satisfies a Sobolev inequality. We show that we can use these estimates to demonstrate that the set of harmonic forms of polynomial growth over M is finite dimensional, under sufficient curvature conditions. We also study in greater detail the dimension of the space of bounded harmonic forms on coverings of compact manifolds.

Coherent states attached to the spectrum of the Bochner Laplacian for the Hopf fibration

We construct coherent states attached to spectrum of the Bochner Laplacian for the Hopf fibration. This enables us to obtain a closed form for the expression of the coherent states attached to Landau levels on the Riemann sphere. We also obtain an identity for certain orthogonal polynomials.

Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians

In this paper, we prove a combinatorial rule describing the restriction of any irreducible representation of U ( n + m ) to the subgroup U ( n ) × U ( m ) . We also derive similar rules for the reductions from SU ( n + m ) to S ( U ( n ) × U ( m ) ) , and from SU ( n + m ) to SU ( n ) × SU ( m ) . As applications of these representation-theoretic results, we compute the spectra of the Bochner–Laplacian on powers of the determinant bundle over the complex Grassmannian Gr n ( C n + m ) . The spectrum of the Dirac operator acting on the spin Grassmannian Gr n ( C n + m ) is also partially computed. A further application is given by the determination of the spectrum of the Hodge–Laplacian acting on the space of smooth functions on the unit determinant bundle over Gr n ( C n + m ) .