An Introduction to Conformal Ricci Flow

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur

What the Infrared Behaviour of QCD Vertex Functions in Landau gauge can tell us about Confinement

The infrared behaviour of Landau gauge QCD vertex functions is investigated employing a skeleton expansion of the Dyson-Schwinger and Renormalization Group equations. Results for the ghost-gluon, three-gluon, four-gluon and quark-gluon vertex functions are presented. Positivity violation of the gluon propagator, and thus gluon confinement, is demonstrated. Results of the Dyson-Schwinger equations for a finite volume are compared to corresponding lattice data. It is analytically demonstrated that a linear rising potential between heavy quarks can be generated by infrared singularities in the dressed quark-gluon vertex. The selfconsistent mechanism that generates these singularities necessarily entails the scalar Dirac amplitudes of the full vertex and the quark propagator. These can only be present when chiral symmetry is broken, either explicitly or dynamically.Comment: 13 pages, 13 figures; to appear in the Proceedings of ``X Hadron Physics 2007'', Florianopolis, Brazil, March 26 - 31, 200

Speed Limits in General Relativity

Some standard results on the initial value problem of general relativity in matter are reviewed. These results are applied first to show that in a well defined sense, finite perturbations in the gravitational field travel no faster than light, and second to show that it is impossible to construct a warp drive as considered by Alcubierre (1994) in the absence of exotic matter.Comment: 7 pages; AMS-LaTeX; accepted for publication by Classical and Quantum Gravit

Hybridization and spin decoherence in heavy-hole quantum dots

We theoretically investigate the spin dynamics of a heavy hole confined to an unstrained III-V semiconductor quantum dot and interacting with a narrowed nuclear-spin bath. We show that band hybridization leads to an exponential decay of hole-spin superpositions due to hyperfine-mediated nuclear pair flips, and that the accordant single-hole-spin decoherence time T2 can be tuned over many orders of magnitude by changing external parameters. In particular, we show that, under experimentally accessible conditions, it is possible to suppress hyperfine-mediated nuclear-pair-flip processes so strongly that hole-spin quantum dots may be operated beyond the `ultimate limitation' set by the hyperfine interaction which is present in other spin-qubit candidate systems.Comment: 7 pages, 3 figure

Sub-10 nm colloidal lithography for integrated spin-photo-electronic devices

Colloidal lithography [1] is how patterns are reproduced in a variety of natural systems and is used more and more as an efficient fabrication tool in bio-, opto-, and nano-technology. Nanoparticles in the colloid are made to form a mask on a given material surface, which can then be transferred via etching into nano-structures of various sizes, shapes, and patterns [2,3]. Such nanostructures can be used in biology for detecting proteins [4] and DNA [5,6], for producing artificial crystals in photonics [7,8] and GHz oscillators in spin-electronics [9-14]. Scaling of colloidal patterning down to 10-nm and below, dimensions comparable or smaller than the main relaxation lengths in the relevant materials, including metals, is expected to enable a variety of new ballistic transport and photonic devices, such as spin-flip THz lasers [15]. In this work we extend the practice of colloidal lithography to producing large-area, near-ballistic-injection, sub-10 nm point-contact arrays and demonstrate their integration in to spin-photo-electronic devices.Comment: 15 pages, 5 figure

A Search for Lost Planets in the Kepler Multi-planet Systems and the Discovery of the Long-period, Neptune-sized Exoplanet Kepler-150 f

The vast majority of the 4700 confirmed planets and planet candidates discovered by the Kepler mission were first found by the Kepler pipeline. In the pipeline, after a transit signal is found, all data points associated with those transits are removed, creating a "Swiss cheese"-like light curve full of holes, which is then used for subsequent transit searches. These holes could render an additional planet undetectable (or "lost"). We examine a sample of 114 stars with $3+$ confirmed planets to evaluate the effect of this "Swiss cheesing". A simulation determines that the probability that a transiting planet is lost due to the transit masking is low, but non-negligible, reaching a plateau at $\sim3.3\%$ lost in the period range of $P=400-500$ days. We then model all planet transits and subtract out the transit signals for each star, restoring the in-transit data points, and use the Kepler pipeline to search the transit-subtracted (i.e., transit-cleaned) light curves. However, the pipeline did not discover any credible new transit signals. This demonstrates the validity and robustness of the Kepler pipeline's choice to use transit masking over transit subtraction. However, a follow-up visual search through all the transit-subtracted data, which allows for easier visual identification of new transits, revealed the existence of a new, Neptune-sized exoplanet (Kepler-150 f) and a potential single transit of a likely false positive (Kepler-208). Kepler-150 f ($P=637.2$ days, $R_{\rm{P}}=3.64^{+0.52}_{-0.39}$ R$_{\oplus}$) is confirmed with $>99.998\%$ confidence using a combination of the planet multiplicity argument, a false positive probability analysis, and a transit duration analysis.Comment: 11 pages, 5 figures, 2 tables. Accepted into A

QCD Propagators at non-vanishing temperatures

We investigate the behaviour of the gluon and ghost propagators, especially their infrared properties, at non-vanishing temperatures. To this end we solve their Dyson-Schwinger equations on a torus and find an infrared enhanced ghost propagator and an infrared vanishing gluon propagator.Comment: 2 pages, 2 figures; talk given by B.G. at the Erice summer school on Nuclear Physics, Sept. 16 -- 24, 2003, Erice, Ital

Perturbations of Spatially Closed Bianchi III Spacetimes

Motivated by the recent interest in dynamical properties of topologically nontrivial spacetimes, we study linear perturbations of spatially closed Bianchi III vacuum spacetimes, whose spatial topology is the direct product of a higher genus surface and the circle. We first develop necessary mode functions, vectors, and tensors, and then perform separations of (perturbation) variables. The perturbation equations decouple in a way that is similar to but a generalization of those of the Regge--Wheeler spherically symmetric case. We further achieve a decoupling of each set of perturbation equations into gauge-dependent and independent parts, by which we obtain wave equations for the gauge-invariant variables. We then discuss choices of gauge and stability properties. Details of the compactification of Bianchi III manifolds and spacetimes are presented in an appendix. In the other appendices we study scalar field and electromagnetic equations on the same background to compare asymptotic properties.Comment: 61 pages, 1 figure, final version with minor corrections, to appear in Class. Quant. Gravi

Testing of pear trees on their own roots in comparison with important used rootstocks under organic farming conditions with special regard to fire blight (E. amylovora)

Pear trees on their own roots are tested in comparison to grafted trees in growth and yield characteristics and with special regard to the tolerance to diseases, above all fire blight (Erwinia amylovora). In spring 2004 15 randomized trees of the cultivar 'Williams' from three variants (self rooted in vitro, self rooted long cuttings, grafted on Quince A) were planted in a pear orchard, which was heavily infected with fire blight (Erwinia amylovora) the previous years. The trees were left untreated. Growth and yield characteristics, plant diseases and tree losses were observed. After four years the in vitro self rooted trees were significantly more vigorous in growth than those grafted on quince A. The self rooted long cuttings were comparable in growth with grafts on quince, but showed high tree losses probably due to frost damages in the first winter one year after planting. However no infections with Erwinia amylovora could be observed so far. In a field trial with more cultivars and rootstock variants planted in 2006 at two organically managed sites more significant effects are expected in the next years