Spectral rigidity and discreteness of 2233-groups

In this paper we describe methods for dealing with the trace spectrum of a subgroup of PSL(2, $\mathbb{R}$) generated by four elliptic elements α, β, γ, δ of respective orders 2, 2, 3, 3, satisfying αβγδ = 1. We give a parametrization and a fundamental domain in the parameter space of such groups. Furthermore we construct an algorithm that decides whether or not a given group is discrete and which moves the discrete groups into the fundamental domain. Our main result is that any two discrete such groups are isospectral if and only if they are conjugate in (2, $\mathbb{R}$). In the Appendix we consider pairs of subgroups of (2, $\mathbb{R}$) that arise from non-conjugate maximal orders in a quaternion algebra over a number field. We show that for the isospectrality of such pairs there is a peculiar exception in the case where the groups contain elements of both orders 2 and

Optimal control of geodesics in Riemannian manifolds

Summary: We present a method based on an optimal control technique for numerical computations of geodesic paths between two fixed points of a Riemannian manifold under the assumption of existence. In this method, the control variable is the tangent vector to the geodesic we are looking for. Defining a cost function corresponding to the requested control, we explain how to derive the optimal control algorithm by the use of an adjoint state method for the calculation of the gradient of that cost function. We then give a geometrical interpretation of the adjoint state. After having introduced the discrete optimal control algorithm, we show an application to wooden roof design

Trace coordinates of Teichmüller space of Riemann surfaces of signature (0,4)(0,4)

Summary: We explicitly give $calT$, the Teichmüller space of four-holed spheres (which we call $X$ pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on $calT$ which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than $2operatornamearccosh(2)$ intersect at most once; two closed geodesics of lengths smaller than $2operatornamearccosh(3)$ are both non-dividing or intersect at most once

The geometry and spectrum of the one holed torus

The authors examine the space of Riemann surfaces of signature (1,1) with metric of curvature -1 and geodesic boundary. They solve explicitly the moduli problem in this case and show furthermore that two surfaces of this type having the same length spectrum (this referring to smooth closed geodesics including the boundary) are isometric. They announce the same type of result for genus two surfaces without boundary. The problem whether the analogous assertion holds for the spectrum of the Laplacian with respect to the Neumann or Dirichlet conditions is open

Some planar isospectral domains

The authors give a number of examples of two noncongruent isospectral domains in the plane and a particularly simple method of proof. One of their examples is a pair of domains that are not only isospectral but homophonic, i.e. each domain has a distinguished point such that the corresponding normalized Dirichlet eigenfunctions take equal values at the distinguished points. They interprete this to mean that if the corresponding ``drums'' are struck at these special points, then they ``sound the same'' in the very strong sense that every frequency will be excited to the same intensity for each. This shows that one really cannot hear the shape of a drum