Monopole metrics and the orbifold Yamabe problem

Jeff A. Viaclovsky[1]

  • [1] University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706 (USA)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 7, page 2503-2543
  • ISSN: 0373-0956

Abstract

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We consider the self-dual conformal classes on n # ℂℙ 2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3 -space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkähler ALE space in dimension four.

How to cite

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Viaclovsky, Jeff A.. "Monopole metrics and the orbifold Yamabe problem." Annales de l’institut Fourier 60.7 (2010): 2503-2543. <http://eudml.org/doc/116344>.

@article{Viaclovsky2010,
abstract = {We consider the self-dual conformal classes on $n \# \mathbb\{CP\}^2$ discovered by LeBrun. These depend upon a choice of $n$ points in hyperbolic $3$-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkähler ALE space in dimension four.},
affiliation = {University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706 (USA)},
author = {Viaclovsky, Jeff A.},
journal = {Annales de l’institut Fourier},
keywords = {Monopole Metrics; Orbifold Yamabe Problem; monopole metrics; orbifold Yamabe problem; conformal classes; Yamabe minimizers},
language = {eng},
number = {7},
pages = {2503-2543},
publisher = {Association des Annales de l’institut Fourier},
title = {Monopole metrics and the orbifold Yamabe problem},
url = {http://eudml.org/doc/116344},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Viaclovsky, Jeff A.
TI - Monopole metrics and the orbifold Yamabe problem
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2503
EP - 2543
AB - We consider the self-dual conformal classes on $n \# \mathbb{CP}^2$ discovered by LeBrun. These depend upon a choice of $n$ points in hyperbolic $3$-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the conformal class of a conformally compactified non-flat hyperkähler ALE space in dimension four.
LA - eng
KW - Monopole Metrics; Orbifold Yamabe Problem; monopole metrics; orbifold Yamabe problem; conformal classes; Yamabe minimizers
UR - http://eudml.org/doc/116344
ER -

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