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The resonant boundary $Q$-curvature problem and boundary-weighted barycenters by Ahmedou, Mohameden; Kallel, Sadok; Ndiaye, Cheikh Birahim is a scholarly article available to read on EtoBox.

What is The resonant boundary $Q$-curvature problem and boundary-weighted barycenters about?

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric problem is equivalent to solving a fourth-order elliptic boundary value problem (BVP) involving the Paneitz operator with boundary conditions of Chang-Qing and Neumann operators. The corresponding BVP has a variational formulation but the corresponding variational problem, in the case under study, is not compact. To overcome such a difficulty we perform a systematic study, \`a la Bahri, of the so called {\it critical points at infinity}, compute their Morse indices, determine their contribution to the difference of topology between the sublevel sets of associated Euler-Lagrange functional and hence extend the full Morse Theory to this noncompact variational problem. To establish Morse inequalities we were led to investigate from the topological viewpoint the space of boundary-weighted barycenters of the underlying manifold, which arise in the description of the topolo

Author

Ahmedou, Mohameden; Kallel, Sadok; Ndiaye, Cheikh Birahim

Published

2016

Language

EN