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The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds  [PDF]
Stefan Ivanov,Dimiter Vassilev
Mathematics , 2015,
Abstract: We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary  [PDF]
YanYan Li,Luc Nguyen
Mathematics , 2009,
Abstract: In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.
Compactness of solutions to the Yamabe problem. II  [PDF]
YanYan Li,Lei Zhang
Mathematics , 2004,
Abstract: We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary  [PDF]
Sorin Dragomir
Mathematics , 2006,
Abstract: We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.
The k-Yamabe problem on CR manifolds  [PDF]
Ezequiel Barbosa,Luiz Gustavo Carneiro,Marcos Montenegro
Mathematics , 2012,
Abstract: We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the CR k-Yamabe problem. We establish that its solutions with null Cotton tensor are critical points of a suitable geometric functional constrained to pseudohermitian structures of unit volume. Thanks to this variational property, we establish a Obata type result for the problem and also compute the infimum of the functional on the CR sphere. Furthermore, we show that this value is an upper bound for the corresponding one on any compact CR manifolds and, assuming the CR Yamabe invariant is positive, we prove that such an upper bound is only attained for compact CR manifolds locally CR-equivalent to the CR sphere. In the Riemannian field, recent advances have been produced in a series of outstanding works.
Yamabe metrics on cylindrical manifolds  [PDF]
Kazuo Akutagawa,Boris Botvinnik
Physics , 2001,
Abstract: We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We affirmatively solve this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cusp singularities. We describe the supremum case, i.e. when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant. Key words: manifolds with cylindrical ends, Yamabe constant/invariant, Yamabe problem, conical metric singularities, cusp metric singularities, Positive Mass Theorem, surgery and Yamabe invariant.
On bifurcation of solutions of the Yamabe problem in product manifolds  [PDF]
L. L. de Lima,P. Piccione,M. Zedda
Mathematics , 2010,
Abstract: We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.
On Yamabe type problems on Riemannian manifolds with boundary  [PDF]
Marco Ghimenti,Anna Maria Micheletti,Angela Pistoia
Mathematics , 2015,
Abstract: Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -\Delta_{g}u+au=0 & \text{ on }M \\ \partial_\nu u+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(M),$ $b\in C^1(\partial M)$, $\nu$ is the outward pointing unit normal to $\partial M $ and $\varepsilon$ is a small positive parameter. We build solutions which blow-up at a point of the boundary as $\varepsilon$ goes to zero. The blowing-up behavior is ruled by the function $b-H_g ,$ where $H_g$ is the boundary mean curvature.
On Bifurcation of Solutions of the Yamabe Problem in Product Manifolds with Minimal Boundary  [PDF]
Elkin Dario Cárdenas Diaz,Ana Cláudia da Silva Moreira
Mathematics , 2015,
Abstract: In this paper, we study multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.
Riemannian manifolds with positive Yamabe invariant and Paneitz operator  [PDF]
Matthew J. Gursky,Fengbo Hang,Yueh-Ju Lin
Mathematics , 2015,
Abstract: For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.
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