Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Aubry, Erwann, and Guillarmou, Colin. "Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity." Journal of the European Mathematical Society 013.4 (2011): 911-957. <http://eudml.org/doc/277372>.

@article{Aubry2011,
abstract = {For odd-dimensional Poincaré–Einstein manifolds $(X^\{n+1\},g)$, we study the set of harmonic $k$-forms (for $k<n/2$) which are $C^m$ (with $m\in \mathbb \{N\}$) on the conformal compactification $\bar\{X\}$ of $X$. This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\bar\{X\},\partial \bar\{X\})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal infinity $(\partial \bar\{X\},[h_0])$. We also relate the set of $C^\{n-2k+1\}(\Lambda ^k(\bar\{X\}))$ forms in the kernel of $d+\delta _g$ to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson–Gover differential operators, including a parallel construction of the generalization of $Q$-curvature for forms.},
author = {Aubry, Erwann, Guillarmou, Colin},
journal = {Journal of the European Mathematical Society},
keywords = {conformal harmonic forms; Branson–Gover operator; harmonic forms; Poincaré-Einstein manifold; harmonic form; Branson-Gover differential operator; -curvature},
language = {eng},
number = {4},
pages = {911-957},
publisher = {European Mathematical Society Publishing House},
title = {Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity},
url = {http://eudml.org/doc/277372},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Aubry, Erwann
AU - Guillarmou, Colin
TI - Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 4
SP - 911
EP - 957
AB - For odd-dimensional Poincaré–Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<n/2$) which are $C^m$ (with $m\in \mathbb {N}$) on the conformal compactification $\bar{X}$ of $X$. This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\bar{X},\partial \bar{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_k,G_k)$ on the conformal infinity $(\partial \bar{X},[h_0])$. We also relate the set of $C^{n-2k+1}(\Lambda ^k(\bar{X}))$ forms in the kernel of $d+\delta _g$ to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson–Gover differential operators, including a parallel construction of the generalization of $Q$-curvature for forms.
LA - eng
KW - conformal harmonic forms; Branson–Gover operator; harmonic forms; Poincaré-Einstein manifold; harmonic form; Branson-Gover differential operator; -curvature
UR - http://eudml.org/doc/277372
ER -