Characteristics of Gain-Guided Laser Diodes with Cylindrical-Mirror Cavity

We have developed highly reliable etched-mirror laser diodes using a dry etching method. The lasers without facet-coating have been operating stably over 2500 h under automatic-power control (APC) at a power of 3 mW/facet at 50°C. The gain-guided laser diodes with a cylindrical-mirror cavity (CMC) have coaxial mirrors and a fan-shaped stripe structure. By decreasing the curvature radius of the inner facet or increasing the stripe width of the inner facet, the beam waist parallel to the junction plane can be moved outside of the laser diode, while the beam waist perpendicular to the junction plane stops at the mirror facet. A particular CMC laser has a low astigmatism of 4.1 μm and a low relative intensity of noise (RIN) less than –134 dB/Hz at 4 mW under 0–1% optical feedback without high frequency current superposition.     

The Complexity of 2-Fold Branched Coverings of a 3-Sphere

Complexity is a function which maps a set of 3-manifolds to a set of nonnegative integers. This function has some natural properties (finiteness, additivity, etc.) and it shows, in some sense how complex the manifold is. This function seems important for the classification of 3-manifolds. We evaluate of the complexity of 2-fold branched coverings of a 3-sphere. We present a theoretical estimate and compare it with experimental data.     

Local Convexity on Smooth Manifolds

Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds. This paper is dedicated to the memory of Guido Stampacchia. This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy.     

Numerical computation of viscous profiles for hyperbolic conservation laws

Viscous profiles of shock waves in systems of conservation laws can be viewed as heteroclinic orbits in associated systems of ordinary differential equations (ODE). In the case of overcompressive shock waves, these orbits occur in multi-parameter families. We propose a numerical method to compute families of heteroclinic orbits in general systems of ODE. The key point is a special parameterization of the heteroclinic manifold which can be understood as a generalized phase condition; in the case of shock profiles, this phase condition has a natural interpretation regarding their stability. We prove that our method converges and present numerical results for several systems of conservation laws. These examples include traveling waves for the Navier-Stokes equations for compressible viscous, heat-conductive fluids and for the magnetohydrodynamics equations for viscous, heat-conductive, electrically resistive fluids that correspond to shock wave solutions of the associated ideal models, i.e., the Euler, resp. Lundquist, equations.

     

An Unsolved Problem of Fenchel

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in R^n>+1 aregiven by C³ n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRⁿ⁺¹.     

Stable complete minimal surfaces in hyperkähler manifolds

In this paper we prove that an isometric stable minimal immersion of a complete oriented surface into a hyperkähler 4-manifold is holomorphic with respect to an orthogonal complex structure, if it satisfies a Bernstein-type assumption on the Gauss-lift. This result generalizes a theorem of Micallef for minimal surfaces in the euclidean 4-space. An example found by Atiyah and Hitchin shows that the assumption on the Gauss-lift is necessary.     

The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

Weak Density of Smooth Maps in W 1, 1(M,N) for Non-Abelian π1(N)

We prove that smooth maps are dense in the sense of biting convergence in W ^{1, 1}(M, N) when M and Nare compact Riemannian manifolds and N is closed.