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

Erwann Aubry; Colin Guillarmou

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 4, page 911-957
- ISSN: 1435-9855

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topAubry, 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 -

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