Ricci flow coupled with harmonic map flow

We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a (possibly time-dependent) positive coupling constant alpha. This system can be interpreted as the gradient flow of an energy functional F_alpha which is a modification of Perelman's energy F for the Ricci flow, including the Dirichlet energy for the map phi. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of phi a-priori - without any assumptions on the curvature of the target manifold N - by choosing alpha large enough. Moreover, if alpha is bounded away from zero it suffices to bound the curvature of (M,g(t)) to also obtain control of phi and all its derivatives - a result which is clearly not true for alpha = 0. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an entropy functional W_alpha similar to Perelman's Ricci flow entropy W and of so-called reduced volume functionals. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.Comment: 41 pages, shortened and references updated, final version, to appear in Ann. Sci. \'Ec. Norm. Sup\'e

Gutachten zuhanden der Sozialdemokratischen Fraktion der schweizerischen Bundesversammlung Betreffend «mögliche Entschädigungsforderungen von AKW-Betreibern»

Rechtsgutachten zu nachfolgenden Fragen: Kann die Betreiberin eines Atomkraftwerks (AKW) Entschädigungsansprüche geltend machen, wenn der Verfassungs- oder der Gesetzgeber die Betriebsdauer nach­träg­lich befristet und das Werk die gesetzlich festgelegte maximale Betriebsdauer er­reicht? Im Rahmen der Hauptfrage sind Entschädigungsanspruch und Bemessungs­grund­lage insbesondere für die folgenden Varianten zu prüfen: - dass die Volksinitiative «Für den geordneten Ausstieg aus der Atomenergie (Atom­ausstiegs­initiative)» der Grünen Partei angenommen wird (Abschaltung des AKW Beznau I ein Jahr nach Annahme der Volksinitiative und Festsetzung einer maxi­malen Betriebs­dauer von 45 Jahren für die vier weiteren AKW auf Verfassungs­ebene); - dass das Parlament mit einer Revision des KEG für alle AKW eine maximale Betriebsdauer von 50 Jahren auf Gesetzesebene festlegt; - dass das Parlament mit einer Revision des KEG für alle AKW eine maximale Betriebsdauer von 60 Jahren auf Gesetzesebene festlegt. Das Gutachten kommt zum Schluss, dass sich aus dem Tatbestand einer materiellen Enteignung oder aus dem verfassungsrechtlichen Vertrauensschutz Schadenersatzansprüche ergeben können. Einen Schaden könnte entstehen, wenn die AKW vor dem Ende ihrer technisch-betrieblichen Lebensdauer ausser Betrieb genommen werden müssen. Ein Ersatzanspruch muss von den AKW-Betreiben nachgewiesen werden und kann nur insoweit bestehen, als Schäden kausal auf eine Rechtsänderung zurückzuführen sind

A Compactness Theorem for Complete Ricci Shrinkers

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argumen

Perelman's Entropy Functional at Type I Singularities of the Ricci Flow

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-M\"{u}ller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result relates a limit W-density at a Type I singular point to the entropy of the limit gradient shrinking soliton obtained by blowing-up at this point. In particular, we show that no entropy is lost at infinity during the blow-up process.Comment: 21 pages. Introduction and references update

Anti-Terrorist Operations and the Right to Life - The European Court of Human Rights case law from Gibraltar to Beslan

The ECtHR's case law on the fundamental right to life has begun to develop with its legal assessment of an anti-terrorist operation in Gibraltar. A more recent judgment in a whole series of cases concerns the Beslan hostage rescue. Both prohibitions and obligations for security force operations follow from that case law. In countering current terrorist threats, there can be a dilemma between the duty to rescue victims and the prohibition of killing perpetrators. This work takes an in-depth look at the three aspects of the right to life under Article 2 ECHR: the positive, the negative and the procedural obligation. Based on this, it rst examines the legal basis for police operations in general and for the use of potentially lethal means of restraint in particular. The distinction between a pre-operational, an operational and a post-operational phase allows existing obligations of states to be properly classied. This shows that state obligations exist both far upstream and in the aftermath of actual operations. The case law on the use of potentially lethal force now forms a close-knit standard for Council of Europe states. It is highly relevant to practice. The present work is intended to stimulate further legal contributions and discussions