Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.     

Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian

In this letter, we give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow a smooth metric tends asymptotically to the metric of constant curvature, this leads to a simple proof of the Osgood–Phillips–Sarnak theorem stating that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of the Laplace operator is maximal on the metric of constant curvatute.Mathematical Subject Classifications (2000). 58J52, 53C44.     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

An Asymptotic Expansion for Bloch Functions on Riemann Surfaces of Infinite Genus and Almost Periodicity of the Kadomcev–Petviashvilli Flow

This article describes the solution of the Kadomcev–Petviashvilli equation with C¹⁰ real periodic initial data in terms of an asymptotic expansion of Bloch functions. The Bloch functions are parametrized by the spectral variety of a heat equation (heat curves) with an external potential. The mentioned spectral variety is a Riemann surface of in general infinite genus; the Kadomcev–Petviashvilli flow is represented by a one-parameter-subgroup in the real part of the Jacobi variety of this Riemann surface. It is shown that the KP-I flow with these initial data propagates almost periodically.     

Anomalous scaling in epitaxial growth on vicinal surfaces: meandering and mounding instabilities in a linear growth equation with spatiotemporally correlated noise

We have undertaken an extensive analytical and kinetic Monte Carlo study of the (2+1) dimensional discrete growth model on a vicinal surface. A non-local, phenomenological continuum equation describing surface growth in unstable systems with anomalous scaling is presented. The roughness produced by unstable growth is first studied considering various effects in surface diffusion processes (corresponding to temperature, flux, diffusion anisotropy). We found that the thermally activated roughness is well-described by a generalized Lai–Das Sarma–Villain model with non linear growth continuum equation and uncorrelated noise. The corresponding critical exponents are computed analytically for the first time and show a continuous variation in agreement with simulation results of a solid-on-solid model. However, the roughness related to the meandering instability is found, unexpectedly, to be well described by a linear continuum equation with spatiotemporally correlated noise.     

Growth of (√3 × √3)-Ag and (1 1 1) oriented Ag islands on Ge/Si(1 1 1) surfaces

We have studied the growth of Ag on Ge/Si(1 1 1) substrates. The Ge/Si(1 1 1) substrates were prepared by depositing one monolayer (ML) of Ge on Si(1 1 1)-(7 × 7) surfaces. Following Ge deposition the reflection high energy electron diffraction (RHEED) pattern changed to a (1 × 1) pattern. Ge as well as Ag deposition was carried out at 550 °C. Ag deposition on Ge/Si(1 1 1) substrates up to 10 ML has shown a prominent (√3 × √3)-R30° RHEED pattern along with a streak structure from Ag(1 1 1) surface. Scanning electron microscopy (SEM) shows the formation of Ag islands along with a large fraction of open area, which presumably has the Ag-induced (√3 × √3)-R30° structure on the Ge/Si(1 1 1) surface. X-ray diffraction (XRD) experiments show the presence of only (1 1 1) peak of Ag indicating epitaxial growth of Ag on Ge/Si(1 1 1) surfaces. The possibility of growing a strain-tuned (tensile to compressive) Ag(1 1 1) layer on Ge/Si(1 1 1) substrates is discussed.     

Growth kinetics of metastable (3 3 1) nanofacet on Au and Pt(1 1 0) surfaces

A theoretical epitaxial growth model with realistic barriers for surface diffusion is investigated by means of kinetic Monte Carlo simulations to study the growth modes of metastable (3 3 1) nanofacets on Au and Pt(1 1 0) surfaces. The results show that under experimental atomic fluxes, the (3 3 1) nanofacets grow by 2D nucleation at low temperature in the submonolayer regime. A metastable growth phase diagram that can be useful to experimentalists is presented and looks similar to the one found for the stationary growth of the bcc(0 0 1) surface in the kinetic 6-vertex model.     

Philipp Frank,Richard von Mises,and the Frank-Mises

The theoretical physicist Philipp Frank (1884–1966) and the applied mathematician Richard von Mises (1883–1953) both received their university education in Vienna shortly after 1900 and became friends at the latest during the Great War.They were attached to the Vienna Circle of Logical Positivists and wrote an influential two-part work on the differential and integral equations of mechanics and physics, the Frank-Mises, of 1925 and 1927, with its second edition following in 1930 and 1935.This work originated in the lectures that the mathematician Bernhard Riemann (1826–1866) delivered on partial differential equations and their applications to physical questions at the University of G?ttingen between 1854 and 1862, which were edited and published posthumously in1869 by the physicist Karl Hattendorff (1834–1882).The immediate precursor of the Frank-Mises, however, was the extensive revision of Hattendorff’s edition of Riemann’s lectures that the mathematician Heinrich Weber (1842–1913) published in two volumes, the Riemann-Weber, of 1900 and 1901, with its second edition following in 1910 and 1912. I trace this historical lineage, explore the nature and contents of the Frank-Mises, and discuss its complementary relationship to the first volume of the text that the mathematicians Richard Courant (1888–1972) and David Hilbert (1862–1943) published on the methods of mathematical physics in 1924, the Courant-Hilbert,which, when it and its second volume of 1937 were translated into English and extensively revised in 1953 and 1961, eclipsed the classic Frank-Mises.