Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> a compact Riemannian <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of <inline-formula><tex-math id="M3">\begin{document}$ g $\end{document}</tex-math></inline-formula> there are scalar-flat metrics that have <inline-formula><tex-math id="M4">\begin{document}$ \partial M $\end{document}</tex-math></inline-formula> as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term <inline-formula><tex-math id="M5">\begin{document}$ h_{g} $\end{document}</tex-math></inline-formula> with a negative smooth function <inline-formula><tex-math id="M6">\begin{document}$ \alpha, $\end{document}</tex-math></inline-formula> the set of solutions of Yamabe problem is still a compact set.

Locations

• Discrete and Continuous Dynamical Systems - S - View - PDF
• arXiv (Cornell University) - View - PDF