Topological restrictions for circle actions and harmonic morphisms

Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is nonnegative and even. In particular, $M^m$ has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example $CP^{n}$, $K3$ surfaces, $S^{2n}\times P_g$ ($n\geq2$) where $P_g$ is the closed surface of genus $g\geq2$ can never be the domain of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and $\phi:(M^4,g)\to(N^3,h)$ a non-constant harmonic morphism. Suppose that one of the following assertions holds: (1) $(M^4,g)$ is half-conformally flat and its scalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat, (3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemannian coverings, $\phi$ is the canonical projection $T^4\to T^3$ between flat tori.Comment: 18 pages; Minor corrections to Proposition 3.1 and small changes in Theorem 2.8, proof of Theorem 3.3 and Remark 3.

Surfaces

Surfaces is a work for two percussionists in four movements. It uses a variety of small percussion instruments, keyboard percussion, and two specially built kalimbas. This document contains both the score and an analysis of the piece that covers instrument choice, performance practice, form and compositional techniques employe

Potential Driven Non-Reactive Phase Transitions of Ordered Porphyrin Molecules on Iodine-Modified Au(100): An Electrochemical Scanning Tunneling Microscopy (EC-STM) Study

The modelling of long-range ordered nanostructures is still a major issue for the scientific community. In this work, the self-assembly of redox-active tetra(N-methyl-4-pyridyl)-porphyrin cations (H2TMPyP) on an iodine-modified Au(100) electrode surface has been studied by means of Cyclic Voltammetry (CV) and in-situ Electrochemical Scanning Tunneling Microscopy (EC-STM) with submolecular resolution. While the CV measurements enable conclusions about the charge state of the organic species, in particular, the potentio-dynamic in situ STM results provide new insights into the self-assembly phenomena at the solid-liquid interface. In this work, we concentrate on the regime of positive electrode potentials in which the adsorbed molecules are not reduced yet. In this potential regime, the spontaneous adsorption of the H2TMPyP molecules on the anion precovered surface yields the formation of up to five different potential-dependent long-range ordered porphyrin phases. Potentio-dynamic STM measurements, as a function of the applied electrode potential, show that the existing ordered phases are the result of a combination of van der Waals and electrostatic interactions

Delaunay Surfaces

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads to simple expressions for the mean and Gaussian curvatures of the surfaces as well as the construction of new surfaces.Comment: 16 pages, 11 figure

Ricci surfaces

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point $x$ of a Ricci surface has a neighborhood which embeds isometrically in $\mathbb{R}^3$ as a minimal surface, provided $K(x)<0$. We prove this result in full generality by showing that Ricci surfaces can be locally isometrically embedded either minimally in $\mathbb{R}^3$ or maximally in $\mathbb{R}^{2,1}$, including near points of vanishing curvature. We then develop the theory of closed Ricci surfaces, possibly with conical singularities, and construct classes of examples in all genera $g\geq 2$.Comment: 27 pages; final version, to appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienz