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Graham Smith

Publications and source records attributed to Graham Smith.

At least 37 records · Page 2Linked to original sources
arXiv2019

Earthquakes and graftings of hyperbolic surface laminations

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive what we believe to be a new formula for the derivative of the length of a simple closed geodesic with respect to the action of grafting. This formula complements those derived by McMullen in [23], in terms of the Weil-Petersson metric, and by Wolpert in [33], for the case of earthquakes.

math.DG↗
arXiv2019

A note on invariant constant curvature immersions in Minkowski space

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a unique affine deformation $Γ:=(ρ,τ)$ of a Fuchsian group such that $(S,g_+)$ and $(S, g_-)$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $Γ$ of an open subset $Ω$ of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Min\-kow\-ski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which $(S, g_+)$ and $(S, g_-)$ are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.

math.DG↗
arXiv2019

Development of a High Rate Front-end ASIC for X-ray Spectroscopy and Diffraction Applications

We developed a new front-end application specific integrated circuit (ASIC) for the upgrade of the Maia x-ray microprobe. The ASIC instruments 32 configurable front-end channels that perform either positive or negative charge amplification, pulse shaping, peak amplitude and time extraction along with buffered analog storage. At a gain of 3.6 V/fC, 1 $μ$s peaking time and a temperature of 248 K, an electronic resolution of 13- and 10 electrons rms was measured with and without a SDD sensor respectively. A spectral resolution of 170 eV FWHM at 5.9 keV was obtained with an $^{55}$Fe source. The channel linearity was better than $\pm$ 1 % with rate capabilities up to 40 kcps. The ASIC was fabricated in a commercial 250 nm process with a footprint of 6.3 mm x 3.9 mm and dissipates 167 mW of static power.

physics.ins-det↗
arXiv2017

Constant scalar curvature hypersurfaces in $(3+1)$-dimensional GHMC Minkowski spacetimes

We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constant scalar curvature. Furthermore, this time function is a smooth submersion.

math.DG↗
arXiv2017

On the Morse index of higher-dimensional free boundary minimal catenoids

For all $n$, we define the $n$-dimensional critical catenoid $M_n$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $\Bbb{R}^{n+1}$. We show that the Morse index $\text{MI}(n)$ of $M_n$ satisfies the following asymptotic estimate as $n$ tends to infinity. $$ \lim_{n\rightarrow+\infty}\frac{\text{Log}(\text{MI}(n))}{\sqrt{n}\text{Log}(\sqrt{n})} = 1. $$ We also study the numerical problem, providing exact values for the Morse index for $n=2,\cdots,100$, together with qualitative studies of $\text{MI}(n)$ and related geometric quantities for large values of $n$.

math.DG↗
arXiv2016

Degree Theory of Immersed Hypersurfaces

We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature; extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-χ(M)$, where $χ(M)$ is the Euler characteristic of $M$.

math.DG↗
arXiv2016

Group actions and scattering problems in Teichmüller theory

In recent years, Teichmüller theory, which is the study of moduli spaces of marked Riemann surfaces, has come to be considered more and more from the point of view of actions of surface groups inside certain semi-simple Lie groups. In particular, we consider the case where the Lie groups in question have symmetric spaces which are lorentzian spacetimes. Indeed, this can be considered as the starting point of Mess' seminal work, which led to the development of new and strikingly simpler proofs of many results of Teichmüller theory by considering them in terms of geometric objects inside these symmetric spaces. Our aim is to provide a brief and straightforward introduction to this approach, whilst developing what we consider to be a useful mental framework for organising known results and open problems.

math.DG↗
arXiv2016

Eternal forced mean curvature flows III - Morse homology

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing term tends to infinity in a certain manner. This result allows the Morse homology to be explicitely calculated, and will permit us to show in forthcoming work that, for a large family of smooth positive functions, $F$, defined over a $(d+1)$-dimensional flat torus, there exist at least $2^{d+1}$ distinct, locally strictly convex, Alexandrov-embedded hyperspheres of mean curvature prescribed at every point by $F$.

math.DG↗
arXiv2015

Eternal Forced Mean Curvature Flows II - Existence

We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.

math.DG↗
arXiv2014

On an Enneper-Weierstrass-type representation of constant Gaussian curvature surfaces in $3$-dimensional hyperbolic space

For all $k\in]0,1[$, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in $3$-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to $k$. We show, furthermore, that this bijection restricts to a homeomorphism over each stratum of the space of ramified coverings of the sphere.

math.DG↗
arXiv2014

The Plateau problem for convex curvature functions

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated.

math.DG↗
arXiv2013

Free boundary minimal annuli in convex three-manifolds

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.

math.DG↗
arXiv2012

Future Science Prospects for AMI

The Arcminute Microkelvin Imager (AMI) is a telescope specifically designed for high sensitivity measurements of low-surface-brightness features at cm-wavelength and has unique, important capabilities. It consists of two interferometer arrays operating over 13.5-18 GHz that image structures on scales of 0.5-10 arcmin with very low systematics. The Small Array (AMI-SA; ten 3.7-m antennas) couples very well to Sunyaev-Zel'dovich features from galaxy clusters and to many Galactic features. The Large Array (AMI-LA; eight 13-m antennas) has a collecting area ten times that of the AMI-SA and longer baselines, crucially allowing the removal of the effects of confusing radio point sources from regions of low surface-brightness, extended emission. Moreover AMI provides fast, deep object surveying and allows monitoring of large numbers of objects. In this White Paper we review the new science - both Galactic and extragalactic - already achieved with AMI and outline the prospects for much more.

astro-ph.IM↗