Special Lagrangians, stable bundles and mean curvature flow

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-H\"older-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.Comment: 36 pages, 4 figures. Minor referee's correction

Understanding the effects of geometry and rotation on pulsar intensity profiles

We have developed a method to compute the possible distribution of radio emission regions in a typical pulsar magnetosphere, taking into account the viewing geometry and rotational effects of the neutron star. Our method can estimate the emission altitude and the radius of curvature of particle trajectory as a function of rotation phase for a given inclination angle, impact angle, spin-period, Lorentz factor, field line constant and the observation frequency. Further, using curvature radiation as the basic emission mechanism, we simulate the radio intensity profiles that would be observed from a given distribution of emission regions, for different values of radio frequency and Lorentz factor. We show clearly that rotation effects can introduce significant asymmetries into the observed radio profiles. We investigate the dependency of profile features on various pulsar parameters. We find that the radiation from a given ring of field lines can be seen over a large range of pulse longitudes, originating at different altitudes, with varying spectral intensity. Preferred heights of emission along discrete sets of field lines are required to reproduce realistic pulsar profiles, and we illustrate this for a known pulsar. Finally, we show how our model provides feasible explanations for the origin of core emission, and also for one-sided cones which have been observed in some pulsars.Comment: 21 pages, 11 figures, accepted for publication in MNRA

Superconductivity and Quantum Phase Transitions in Weak Itinerant Ferromagnets

It is argued that the phase transition in low-T_c clean itinerant ferromagnets is generically of first order, due to correlation effects that lead to a nonanalytic term in the free energy. A tricritical point separates the line of first order transitions from Heisenberg critical behavior at higher temperatures. Sufficiently strong quenched disorder suppresses the first order transition via the appearance of a critical endpoint. A semi-quantitative discussion is given in terms of recent experiments on MnSi and UGe_2. It is then shown that the critical temperature for spin-triplet, p-wave superconductivity mediated by spin fluctuations is generically much higher in a Heisenberg ferromagnetic phase than in a paramagnetic one, due to the coupling of magnons to the longitudinal magnetic susceptibility. This qualitatively explains the phase diagram recently observed in UGe_2 and ZrZn_2.Comment: 10 pp., LaTeX, 5 ps figs., requires World Scientific style files (included), Invited contribution to MB1

Quantum critical behavior of clean itinerant ferromagnets

We consider the quantum ferromagnetic transition at zero temperature in clean itinerant electron systems. We find that the Landau-Ginzburg-Wilson order parameter field theory breaks down since the electron-electron interaction leads to singular coupling constants in the Landau-Ginzburg-Wilson functional. These couplings generate an effective long-range interaction between the spin or order parameter fluctuations of the form 1/r^{2d-1}, with d the spatial dimension. This leads to unusual scaling behavior at the quantum critical point in 1 < d\leq 3, which we determine exactly. We also discuss the quantum-to-classical crossover at small but finite temperatures, which is characterized by the appearance of multiple temperature scales. A comparison with recent results on disordered itinerant ferromagnets is given.Comment: 13 pp., REVTeX, psfig, 3 eps figs, final version as publishe

Second-Order Convergence of a Projection Scheme for the Incompressible Navier–Stokes Equations with Boundaries

A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and the pressure is chosen so that the computed velocity satisfies a discrete divergence-free condition. This choice for the pressure and the particular way that the discrete divergence is calculated near the boundary permit the error in the pressure to be controlled and the second-order convergence in the velocity and the pressure to the exact solution to be shown. Some simplifications in the calculation of the pressure in the case without boundaries are also discussed

Influence of Generic Scale Invariance on Classical and Quantum Phase Transitions

This review discusses a paradigm that has become of increasing importance in the theory of quantum phase transitions; namely, the coupling of the order-parameter fluctuations to other soft modes, and the resulting impossibility of constructing a simple Landau-Ginzburg-Wilson theory in terms of the order parameter only. The soft modes in question are manifestations of generic scale invariance, i.e., the appearance of long-range order in whole regions in the phase diagram. The concept of generic scale invariance, and its influence on critical behavior, is explained using various examples, both classical and quantum mechanical. The peculiarities of quantum phase transitions are discussed, with emphasis on the fact that they are more susceptible to the effects of generic scale invariance than their classical counterparts. Explicit examples include: the quantum ferromagnetic transition in metals, with or without quenched disorder; the metal-superconductor transition at zero temperature; and the quantum antiferromagnetic transition. Analogies with classical phase transitions in liquid crystals and classical fluids are pointed out, and a unifying conceptual framework is developed for all transitions that are influenced by generic scale invariance.Comment: 55pp, 25 eps figs; final version, to appear in Rev Mod Phy

Axisymmetric stationary solutions with arbitrary multipole moments

In this paper, the problem of finding an axisymmetric stationary spacetime from a specified set of multipole moments, is studied. The condition on the multipole moments, for existence of a solution, is formulated as a convergence condition on a power series formed from the multipole moments. The methods in this paper can also be used to give approximate solutions to any order as well as estimates on each term of the resulting power series.Comment: 12 page