Multiplicity results for the prescribed scalar curvature on low spheres

Mohamed Ben Ayed; Mohameden Ould Ahmedou

Displaying similar documents to “Multiplicity results for the prescribed scalar curvature on low spheres”

Curvature measures, normal cycles and asymptotic cones

Xiang Sun, Jean-Marie Morvan (2013)

Actes des rencontres du CIRM

Similarity:

The purpose of this article is to give an overview of the theory of the and to show how to use it to define a on singular surfaces embedded in an (oriented) Euclidean space $𝔼^{3}$ . In particular, we will introduce the notion of associated to a Borel subset of $𝔼^{3}$ , generalizing the defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the $3$ -dimensional Euclidean space $𝔼^{3}$ . The coherence of the theory lies in a convergence theorem: If a...

A new kind of augmentation of filtrations

Joseph Najnudel, Ashkan Nikeghbali (2011)

ESAIM: Probability and Statistics

Similarity:

Let (Ω, $ℱ$ , ( $ℱ_{t}$ ), $ℙ$ ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $ℱ_{\infty}$ (the-algebra generated by ( $ℱ_{t}$ )) a coherent family of probability measures ( $ℚ_{t}$ ) indexed by , each of them being defined on $ℱ_{t}$ . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative...

Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez, Jean-Jacques Risler (2010)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let $Δ \subset 𝐑^{2}$ be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on $Δ$ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by $Δ$ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C, 0)$ . We introduce the class ofsmoothings of $(C, 0)$ by...

A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$ -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

Similarity:

In this paper we consider Riemannian manifolds $(M^{n}, g)$ of dimension $n \geq 5$ , with semi-positive $Q$ -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$ -curvature. Modifying the test function construction of Esposito-Robert,...

A proof of Reidemeister-Singer’s theorem by Cerf’s methods

François Laudenbach (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

Heegaard splittings and Heegaard diagrams of a closed 3-manifold $M$ are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on $M$ . We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when $dim M > 2$ . The main tool that we introduce is an which could be useful elsewhere.

Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

Similarity:

Let $𝔽_{}^{}$ F be a filtration andbe a random time. Let $𝔾_{}^{}$ G be the progressive enlargement of $𝔽_{}^{}$ F with. We study the following formula, called the optional splitting formula: For any $𝔾_{}^{}$ G-optional process, there exists an $𝔽_{}^{}$ F-optional process and a function defined on [0∞] × (ℝ × ) being $ℬ [0, \infty] \otimes 𝒪 (𝔽_{}^{})$ ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y = Y^{'} 1_{[0, τ)}^{} + Y^{''} (τ) 1_{[τ, \infty)}^{} .$ Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random times ...

Cycle structure of percolation on high-dimensional tori

Remco van der Hofstad, Artëm Sapozhnikov (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdős–Rényi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the of the largest...

A new characterization of the sphere in $R^{3}$

Thomas Hasanis (1980)

Annales Polonici Mathematici

Similarity:

Let M be a closed connected surface in $R^{3}$ with positive Gaussian curvature K and let $K_{I} I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I} I = c \sqrt H K^{r}$ , for some constants c and r, where H is the mean curvature of M.

The resolution of the bounded $L^{2}$ curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

This paper reports on the recent proof of the bounded $L^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^{2}$ -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

Scaling laws for non-euclidean plates and the $W^{2, 2}$ isometric immersions of riemannian metrics

Marta Lewicka, Mohammad Reza Pakzad (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its -convergence under...

A priori bounds for some infinitely renormalizable quadratics: II. Decorations

Jeremy Kahn, Mikhail Lyubich (2008)

Annales scientifiques de l'École Normale Supérieure

Similarity:

A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove bounds. They imply local connectivity of the corresponding Julia...

Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

In this article, we study the approximation of a probability measure $μ$ on $ℝ^{d}$ by its empirical measure ${\hat{μ}}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$ th moment Wasserstein metric. In the case where $2 p l t; d$ , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

Ricci flow coupled with harmonic map flow

Reto Müller (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $φ$ from $M$ to some closed target manifold $N$ , $\frac{\partial}{\partial t} g = - 2 Rc + 2 α \nabla φ \otimes \nabla φ, \frac{\partial}{\partial t} φ = τ_{g} φ,$ where $α$ is a (possibly time-dependent) positive coupling constant. 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 $φ$ a-priori by choosing $α$ large enough. Moreover, it suffices to bound the curvature...

Global pinching theorems for minimal submanifolds in spheres

Kairen Cai (2003)

Colloquium Mathematicae

Similarity:

Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n + p} (1)$ . By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${| | σ | |}_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${| | σ | |}_{n / 2} < C$ , then M is a minimal submanifold in the sphere $S^{n + p - 1} (1 + H ²)$ with sectional...

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Benjamin Charlier (2013)

ESAIM: Probability and Statistics

Similarity:

Let ( $𝕊^{1}, d_{𝕊^{1}}$ S 1 , d S 1 ) be the unit circle in ℝ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure to admit a well defined Fréchet mean on ( $𝕊^{1}, d_{𝕊^{1}}$ S 1 , d S 1 ). We derive a new sufficient condition of existence() with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

Digital shapes, digital boundaries and rigid transformations: A topological discussion

Yukiko Kenmochi, Phuc Ngo, Nicolas Passat, Hugues Talbot (2013)

Actes des rencontres du CIRM

Similarity:

Curvature is a continuous and infinitesimal notion. These properties induce geometrical difficulties in digital frameworks, and the following question is naturally asked: “How to define and compute curvatures of digital shapes?” In fact, not only geometrical but also topological difficulties are also induced in digital frameworks. The – deeper – question thus arises: “Can we still define and compute curvatures?” This latter question, that is relevant in the context of digitization, ,...

The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra

Vladimir V. Bavula (2010)

Bulletin de la Société Mathématique de France

Similarity:

There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian...

Waring’s problem for Beatty sequences and a local to global principle

William D. Banks, Ahmet M. Güloğlu, Robert C. Vaughan (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We investigate in various ways the representation of a large natural number $N$ as a sum of $s$ positive $k$ -th powers of numbers from a fixed Beatty sequence. , a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser.