A Monge-Ampère equation in conformal geometry

Matthew J. Gursky

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 2, page 241-270
  • ISSN: 0391-173X

Abstract

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We consider the Monge-Ampère-type equation det ( A + λ g ) = const . , where A is the Schouten tensor of a conformally related metric and λ > 0 is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

How to cite

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Gursky, Matthew J.. "A Monge-Ampère equation in conformal geometry." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.2 (2008): 241-270. <http://eudml.org/doc/272266>.

@article{Gursky2008,
abstract = {We consider the Monge-Ampère-type equation $\det (A + \lambda g) = \mathrm \{const\}.$, where $A$ is the Schouten tensor of a conformally related metric and $\lambda &gt; 0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.},
author = {Gursky, Matthew J.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Monge-Ampère equation; -Yamabe problem; conformal geometry; nonlinear equations},
language = {eng},
number = {2},
pages = {241-270},
publisher = {Scuola Normale Superiore, Pisa},
title = {A Monge-Ampère equation in conformal geometry},
url = {http://eudml.org/doc/272266},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Gursky, Matthew J.
TI - A Monge-Ampère equation in conformal geometry
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 2
SP - 241
EP - 270
AB - We consider the Monge-Ampère-type equation $\det (A + \lambda g) = \mathrm {const}.$, where $A$ is the Schouten tensor of a conformally related metric and $\lambda &gt; 0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.
LA - eng
KW - Monge-Ampère equation; -Yamabe problem; conformal geometry; nonlinear equations
UR - http://eudml.org/doc/272266
ER -

References

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  10. [10] R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations, (Montecatini Terme, 1987), Lecture Notes in Math., Vol. 1365, Springer, Berlin, 1989, 120–154. Zbl0702.49038MR994021
  11. [11] W.-M. Sheng, N. S. Trudinger and X.-J. Wang, The Yamabe problem for higher order curvatures, J. Differential Geom.77 (2007), 515–553. Zbl1133.53035MR2362323
  12. [12] J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J.101 (2000), 283–316. Zbl0990.53035MR1738176
  13. [13] J. A. Viaclovsky, Conformal geometry and differential equations, to appear in: “Inspired by S. S. Chern: A Memorial Volume in Honor of a Great Mathematician”, P. Griffiths (ed.), Nankai Tracts in Mathematics, Vol. II, World Scientific, 2006. Zbl1142.53030MR1764770

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