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Prescribing a fourth order conformal invariant on the standard sphere, part II : blow up analysis and applications

Zindine Djadli, Andrea Malchiodi, Mohameden Ould Ahmedou (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere ( 𝕊 n , h ) . We derive from this analysis some a priori estimates in dimension 5 and 6 . On 𝕊 5 these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On 𝕊 6 we prove the existence of at least one...

Prescribing Q -curvature on higher dimensional spheres

Khalil El Mehdi (2005)

Annales mathématiques Blaise Pascal

We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.

Principe de recollement des équations des contraintes en relativité générale

Julien Cortier (2011/2012)

Séminaire de théorie spectrale et géométrie

La méthode de «  recollement  » permettant de trouver des solutions des équations des contraintes relativistes est décrite. En particulier, on expose la méthode de Corvino-Schoen pour construire des familles de solutions sur une variété non-compacte avec géométrie prescrite sur un bout asymptotique, en insistant sur le recollement «  non-localisé  ». Une liste de résultats obtenus par divers auteurs à partir de telles techniques est alors fournie, incluant la question du recollement de métriques...

Problèmes de Yamabe généralisés et ses applications

Yuxin Ge (2006/2007)

Séminaire de théorie spectrale et géométrie

On étudie quelques équations complètement non linéaires issues de la géométrie conforme. Par une méthode de flot géométrique, on prouve l’existence des solutions. En utilisant ce résultat analytique, on obtient un théorème sur la topologie de la variété : soit M une variété riemannienne compacte de dimension 3. S’il existe une metrique g à courbure scalaire strictement positive telle que l’intégrale de la σ 2 -courbure scalaire soit positive, alors M est difféomorphe à un quotient de la sphere.

Refined Kato inequalities in riemannian geometry

Marc Herzlich (2000)

Journées équations aux dérivées partielles

We describe the recent joint work of the author with David M. J. Calderbank and Paul Gauduchon on refined Kato inequalities for sections of vector bundles living in the kernel of natural first-order elliptic operators.

Resolvent Flows for Convex Functionals and p-Harmonic Maps

Kazuhiro Kuwae (2015)

Analysis and Geometry in Metric Spaces

We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT(0)-spaces. The results can be applied to Lp-Wasserstein space over complete p-uniformly convex spaces. As an application,...

Ricci flow coupled with harmonic map flow

Reto Müller (2012)

Annales scientifiques de l'École Normale Supérieure

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 , ∂ ∂ t g = - 2 Rc + 2 α ∇ φ ⊗ ∇ φ , ∂ ∂ 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 of  ( M , g ( t ) ) to also obtain control of ...

Rigidity and L 2 cohomology of hyperbolic manifolds

Gilles Carron (2010)

Annales de l’institut Fourier

When X = Γ ∖ ℍ n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

Rigidity of Einstein manifolds and generalized quasi-Einstein manifolds

Yi Hua Deng, Li Ping Luo, Li Jun Zhou (2015)

Annales Polonici Mathematici

We discuss the rigidity of Einstein manifolds and generalized quasi-Einstein manifolds. We improve a pinching condition used in a theorem on the rigidity of compact Einstein manifolds. Under an additional condition, we confirm a conjecture on the rigidity of compact Einstein manifolds. In addition, we prove that every closed generalized quasi-Einstein manifold is an Einstein manifold provided μ = -1/(n-2), λ ≤ 0 and β ≤ 0.

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