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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.
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.
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.
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.
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.
An existence theorem for the Yamabe problem on manifolds with boundary  [PDF]
S. Brendle,S. Chen
Mathematics , 2009,
Abstract: Let (M,g) be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar's work. Moreover, we reduce the remaining cases to the positive mass theorem.
Fractional conformal Laplacians and fractional Yamabe problems  [PDF]
Maria del Mar Gonzalez,Jie Qing
Mathematics , 2010,
Abstract: Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type maximum principle together with interplays between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extend the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem.
The zero scalar curvature Yamabe problem on noncompact manifolds with boundary  [PDF]
Fernando Schwartz
Mathematics , 2006,
Abstract: Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curvature $f$ on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.
Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometry  [PDF]
Nadine Gro?e
Mathematics , 2010,
Abstract: We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hijazi inequality for the underlying spin manifold.
Compactness and Non-compactness for the Yamabe Problem on Manifolds With Boundary  [PDF]
Marcelo M. Disconzi,Marcus A. Khuri
Mathematics , 2012,
Abstract: We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.
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