No abstract

The evolution of quantum light through linear optical devices can be described by the scattering matrix S of the system. For linear optical systems with m possible modes, the evolution of n input photons is given by a unitary matrix U = ϕ m,M (S), derived from a known homomorphism, ϕ m,M , which depends on the size of the resulting Hilbert space of the possible photon states, M. We present a method to decide whether a given unitary evolution U for n photons in m modes can be achieved with linear optics or not and the inverse transformation ϕ −1 m,M when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

Abstract. Given a complete isometric immersion ϕ : P m −→ N n in an ambient Riemannian manifold N n with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space M n w , we determine a set of conditions on the extrinsic curvatures of P that guarantees that the immersion is proper and that P has finite topology in the line of the results in [24] and [25]. When the ambient manifold is a radially symmetric space, it is shown an inequality between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends which generalizes the classical inequality stated in [1] for complete and minimal submanifolds in R n . We obtain as a corollary the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in the Hyperbolic space together with Bernstein type results for such submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work [12].

Abstract. We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above.

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We study the topology of (properly) immersed complete minimal surfaces P 2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see [12]). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in R n and in H n (b)), based in the isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.2000 Mathematics Subject Classification. Primary 53C20 ; Secondary 53C42, 49Q05.

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.

This article has results of four types. We show that the first eigenvalue λ1(Ω) of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit λ1Then, we study the L 1 (Ω, μ)-moment spectrum of Ω in terms of iterates of the Green operator G, extending the work of McDonald-Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the L 1 (Ω, μ)moment spectrum, generalizing the work of Hurtado-Markvorsen-Palmer. Finally, we study the radial spectrum σ rad (B h (o, r)) of rotationally invariant geodesic balls B h (o, r) of model manifolds. We prove an identity relating the radial eigenvalues of σ rad (B h (o, r)) to an isoperimetric quotient, i.e., 1/λ rad i = V (s)/S(s)ds, V (s) = vol(B h (o, s)) and S(s) = vol(∂B h (o, s)). We then consider a proper minimal surface M ⊂ R 3 and the extrinsic ball Ω = M ∩ B R 3 (o, r). We obtain upper and lower estimates for the series λ −2 i (Ω) in terms of the volume vol(Ω) and the radius r of the extrinsic ball Ω.