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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.
On the Yamabe Problem on contact Riemannian Manifolds  [PDF]
Feifan Wu,Wei Wang
Mathematics , 2015,
Abstract: Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.
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.
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 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.
A Yamabe-type problem on manifolds with boundary
Fernando Coda Marques
Matemáticas : Ense?anza Universitaria , 2007,
Abstract: In this article we give a brief survey on a Yamabe-type problem on manifolds with boundary. Given a compact manifold (Mn, g), with nonempty boundary, the problem consists in finding a conformal metric of zero scalar curvature and constant mean curvature on the boundary
Estimates and Existence Results for a Fully Nonlinear Yamabe Problem on Manifolds with Boundary  [PDF]
Qinian Jin,Aobing Li,YanYan Li
Mathematics , 2006,
Abstract: This paper concerns a fully nonlinear version of the Yamabe problem on manifolds with boundary. We establish some existence results and estimates of solutions.
A combinatorial Yamabe problem on two and three dimensional manifolds  [PDF]
Huabin Ge,Xu Xu
Mathematics , 2015,
Abstract: In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.
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.
On Harnack inequalities and singularities of admissible metrics in the Yamabe problem  [PDF]
Neil S. Trudinger,Xu-Jia Wang
Mathematics , 2005,
Abstract: In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds $(M, g_0)$ of dimension $n\ge 3$. For $n/2 n/2$.
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