Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction

Stabilising Lyme Regis – a strategic approach

Coastal erosion and landslides have been a constant threat to Lyme Regis in West Dorset, UK for over 250 years. By the 1980s, the frequency and scale of coastal erosion and land instability had reached a point whereby the local council realised that a change from the previous ad hoc repair and protection approach was needed to secure the long-term future of the town. An environmental improvements initiative was developed from then onwards to provide a strategic and integrated programme of coast protection and cliff stabilisation measures designed to mitigate the increasing threat of climate change, coastal erosion and landslides, while respecting the site’s unique heritage and environmental interests. This paper outlines the background and principal phases of the project that have been successfully delivered over the period 1990–2015

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

Eternal solutions to a singular diffusion equation with critical gradient absorption

The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular diffusion equation with critical gradient absorption \begin{equation*} \partial_{t} u-\Delta_{p} u+|\nabla u|^{p/2}=0 \quad \;\;\hbox{in}\;\; (0,\infty)\times\real^N \end{equation*} where $2N/(N+1) < p < 2$. Such solutions are shown to exist only if the parameter $\beta$ ranges in a bounded interval $(0,\beta_*]$ which is in sharp contrast with well-known singular diffusion equations such as $\partial_{t}\phi-\Delta_{p} \phi=0$ when $p=2N/(N+1)$ or the porous medium equation $\partial_{t}\phi-\Delta\phi^m=0$ when $m=(N-2)/N$. Moreover, the profile $f(r;\beta)$ decays to zero as $r\to\infty$ in a faster way for $\beta=\beta_*$ than for $\beta\in (0,\beta_*)$ but the algebraic leading order is the same in both cases. In fact, for large $r$, $f(r;\beta_*)$ decays as $r^{-p/(2-p)}$ while $f(r;\beta)$ behaves as $(\log r)^{2/(2-p)} r^{-p/(2-p)}$ when $\beta\in (0,\beta_*)$

Formal matched asymptotics for degenerate Ricci flow neckpinches

Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on $S^m$, for all $m\geq 3$. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit

Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family $\mu_P^{s,t}$ of measures on a space of functions on the two-torus, parametrized by a polynomial $P$ (the Wess-Zumino-Landau-Ginzburg model). The second is a family \mu_\cG^{s,t} of measures on a space \cG of maps from $\P^1$ to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family $\mu_{M,G}^{s,t}$ of measures on the product of a space of connection s on the trivial principal bundle with structure group $G$ on a three-dimensional manifold $M$ with a space of \fg-valued three-forms on $M.$ We show that these measures are positive, and that the measures \mu_\cG^{s,t} are Borel probability measures. As an application we show that formulas arising from expectations in the measures \mu_\cG^{s,1} reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures $\mu_{M,SU(2)}^{s,t},$ where $M$ is a homology three-sphere, will yield the Casson invariant of $M.$Comment: Minor correction

Social marketing and healthy eating : Findings from young people in Greece

This document is the Accepted Manuscript version. The final publication is available at Springer via http://dx.doi.org/10.1007/s12208-013-0112-xGreece has high rates of obesity and non-communicable diseases owing to poor dietary choices. This research provides lessons for social marketing to tackle the severe nutrition-related problems in this country by obtaining insight into the eating behaviour of young adults aged 18–23. Also, the main behavioural theories used to inform the research are critically discussed. The research was conducted in Athens. Nine focus groups with young adults from eight educational institutions were conducted and fifty-nine participants’ views towards eating habits, healthy eating and the factors that affect their food choices were explored. The study found that the participants adopted unhealthier nutritional habits after enrolment. Motivations for healthy eating were good health, appearance and psychological consequences, while barriers included lack of time, fast-food availability and taste, peer pressure, lack of knowledge and lack of family support. Participants reported lack of supportive environments when deciding on food choices. Based on the findings, recommendations about the development of the basic 4Ps of the marketing mix, as well as of a fifth P, for Policy are proposedPeer reviewe

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hölder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs