Terms Of Geometric Data Research Articles

Energy distribution of harmonic 1-forms and Jacobians of Riemann surfaces with a short closed geodesic

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface S where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian variety of S develops into a variety that splits. If the geodesic is nonseparating then the Jacobian degenerates. The aim of this work is to get insight into this process and give estimates in terms of geometric data of both the initial surface S and the final surface, such as its injectivity radius and the lengths of geodesics that form a homology basis. The Jacobians in this paper are represented by Gram period matrices. As an invariant we introduce new families of symplectic matrices that compensate for the lack of full dimensional Gram-period matrices in the noncompact case.

Conformal Patterson–Walker metrics

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.

Heat Flow Out of a Compact Manifold

We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series, and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.

On a theorem of Peter Scott

We quantify Peter Scott’s theorem that surface groups are locally extended residually finite (LERF) in terms of geometric data. In the process, we will quantify another result by Scott that any closed geodesic in a surface lifts to an embedded loop in a finite cover.

Area Quantization of the Parameter Space of Riemann Surfaces in Genus Two

We consider a model of compact Riemann surfaces in genus two represented geometrically by two-parametric hyperbolic octagons with an order п/2 automorphism. We compute the generators of the Fuchsian group and give a real-analytic description of the corresponding Teichm¨uller space parametrized by the Fenchel–Nielsen variables in terms of geometric data. We state the structure of the parameter space by computing the Weil–Petersson (WP) symplectictwo-form and analyzing the isoperimetric orbits. Combining these results, the WP area in the parameter space and the canonical action–angle variables for the orbits are found. Using the ideas from the loop quantum gravity, we apply our formalism to the description of the classical geometrodynamics of Riemann surfaces and the WP area quantization. The results of the paper may be interesting due to their applications to the quantum geometry, chaotic systems, and low-dimensional gravity.

Heat Trace Asymptotics with Singular Weight Functions II

We study the weighted heat trace asymptotics of an operator of Laplace type with mixed boundary conditions where the weight function exhibits radial blowup. We give formulas for the first three boundary terms in the expansion in terms of geometrical data.

Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism, the scattering matrix at low energy may be regarded as an operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud–Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.

Heat trace asymptotics with singular weight functions

We study the weighted heat trace asymptotics of an operator of Laplace type with mixed boundary conditions where the weight function exhibits radial blowup. We give formulas for the first three boundary terms in the expansion in terms of geometrical data.

On the local structure of Dirac manifolds

AbstractWe give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through m. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

On period maps that are open embeddings

For certain complex projective manifolds (such as K3 surfaces and their higher dimensional analogues, the complex symplectic projective manifolds) the period map takes values in a locally symmetric variety of type IV. It is often an open embedding and in such cases it has been observed that the image is the complement of a locally symmetric divisor. We explain that phenomenon and get our hands on the complementary divisor in terms of geometric data.

On the geometry of matrix models for Script N = 1*

We investigate the geometry of the matrix model associated with an N=1 super Yang-Mills theory with three adjoint fields, which is a massive deformation of N=4. We study in particular the Riemann surface underlying solutions with arbitrary number of cuts. We show that an interesting geometrical structure emerges where the Riemann surface is related on-shell to the Donagi-Witten spectral curve. We explicitly identify the quantum field theory resolvents in terms of geometrical data on the surface.

The fundamental solution on manifolds with time-dependent metrics

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.

Semiprime Graded Algebras of Dimension Two

Semiprime, noetherian, connected graded k-algebras R of quadratic growth are described in terms of geometric data. A typical example of such a ring is obtained as follows: Let Y be a projective variety of dimension at most one over the base field k and let E be an OY-order in a finite dimensional semisimple algebra A over K=k(Y). Then, for any automorphism τ of A that restricts to an automorphism σ of Y and any ample, invertible E-bimodule B, Van den Bergh constructs a noetherian, “twisted homogeneous coordinate ring” B=⊕H0(Y,B⊗···⊗Bτn−1). We show that R is noetherian if and only if some Veronese ring R(m) of R has the form k+I, where I is a left ideal of such a ring B and where I=B at each point p∈Y at which σ has finite order. This allows one to give detailed information about the structure of R and its modules.

Noncommutative graded domains with quadratic growth

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.

Invariance Groups in Classical and Quantum Mechanics

Abstract This is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

Sound Propagation in Lamellar Composite Materials and Averaged Elastic Constants

Dispersion relations for the propagation of elastic waves in a lamellar periodic composite are obtained for wavelengths that are long as compared with the intercomponent spacings. This yields exact explicit expressions for the complete set of five independent, elastic constants of the material, in terms of geometrical data and the elastic constants of the constituents. The expressions are valid for any spatially varying elastic moduli and for density-showing periodic structure and mirror symmetry within the elementary cell.

Elastic Constants of Filamentary Composites with Rectangular Symmetry

Dispersion relations for the propagation of elastic waves in filamentary periodic composites with rectangular symmetry are obtained for wavelengths that are long as compared with the intercomponent spacings. This yields averaged elastic constants of the material in terms of geometrical data and the elastic constants of the constituents. General averaging rules are established for all nine independent elastic constants. These rules are valid for any spatially varying elastic moduli and density with rectangular symmetry in the elementary cell. In the special case of a square lattice, the formulas are evaluated in the Wigner-Seitz approximation. For two constituents, this gives explicit expressions for all six independent elastic constants.