Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri

Bulletin de la Société Mathématique de France (2009)

  • Volume: 137, Issue: 1, page 63-91
  • ISSN: 0037-9484

Abstract

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To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M × ( - 1 , 0 ) . We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × ( - 1 , 0 ) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact form θ ), i.e., is an invariant of the contact structure H . The approximately Einstein ACH metric g is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman’s approximately Einstein complete Kähler metric g + on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of g + is in fact a contact invariant. We discuss some implications this may have for CR Q -curvature. The formal power series method of finding g is obstructed at finite order. We show that part of this obstruction is given as a one-form on H * . This is a new result peculiar to the partially integrable setting.

How to cite

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Seshadri, Neil. "Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds." Bulletin de la Société Mathématique de France 137.1 (2009): 63-91. <http://eudml.org/doc/272398>.

@article{Seshadri2009,
abstract = {To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact form $\theta $), i.e., is an invariant of the contact structure $H$. The approximately Einstein ACH metric $g$ is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman’s approximately Einstein complete Kähler metric $g_+$ on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of $g_+$ is in fact a contact invariant. We discuss some implications this may have for CR $Q$-curvature. The formal power series method of finding $g$ is obstructed at finite order. We show that part of this obstruction is given as a one-form on $H^*$. This is a new result peculiar to the partially integrable setting.},
author = {Seshadri, Neil},
journal = {Bulletin de la Société Mathématique de France},
keywords = {ACH metric; approximately Einstein metric; volume renormalization; contact manifold; almost CR structure; CR $Q$-curvature; CR obstruction tensor; CR -curvature},
language = {eng},
number = {1},
pages = {63-91},
publisher = {Société mathématique de France},
title = {Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds},
url = {http://eudml.org/doc/272398},
volume = {137},
year = {2009},
}

TY - JOUR
AU - Seshadri, Neil
TI - Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds
JO - Bulletin de la Société Mathématique de France
PY - 2009
PB - Société mathématique de France
VL - 137
IS - 1
SP - 63
EP - 91
AB - To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact form $\theta $), i.e., is an invariant of the contact structure $H$. The approximately Einstein ACH metric $g$ is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman’s approximately Einstein complete Kähler metric $g_+$ on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of $g_+$ is in fact a contact invariant. We discuss some implications this may have for CR $Q$-curvature. The formal power series method of finding $g$ is obstructed at finite order. We show that part of this obstruction is given as a one-form on $H^*$. This is a new result peculiar to the partially integrable setting.
LA - eng
KW - ACH metric; approximately Einstein metric; volume renormalization; contact manifold; almost CR structure; CR $Q$-curvature; CR obstruction tensor; CR -curvature
UR - http://eudml.org/doc/272398
ER -

References

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