Conformally bending three-manifolds with boundary

Matthew Gursky[1]; Jeffrey Streets[2]; Micah Warren[2]

  • [1] University of Notre Dame Department of Mathematics Notre Dame, IN 46556 (USA)
  • [2] Princeton University Fine Hall Princeton, NJ 08544 (USA)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 7, page 2421-2447
  • ISSN: 0373-0956

Abstract

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Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant. This condition roughly means that the curvature is “almost negative”, in a scale-invariant sense.

How to cite

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Gursky, Matthew, Streets, Jeffrey, and Warren, Micah. "Conformally bending three-manifolds with boundary." Annales de l’institut Fourier 60.7 (2010): 2421-2447. <http://eudml.org/doc/116340>.

@article{Gursky2010,
abstract = {Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant. This condition roughly means that the curvature is “almost negative”, in a scale-invariant sense.},
affiliation = {University of Notre Dame Department of Mathematics Notre Dame, IN 46556 (USA); Princeton University Fine Hall Princeton, NJ 08544 (USA); Princeton University Fine Hall Princeton, NJ 08544 (USA)},
author = {Gursky, Matthew, Streets, Jeffrey, Warren, Micah},
journal = {Annales de l’institut Fourier},
keywords = {Almost negative curvature; conformal filling; fully nonlinear equations; manifold with boundary; conformal bending; scalar curvature; pinching of sectional curvature; almost negative curvature},
language = {eng},
number = {7},
pages = {2421-2447},
publisher = {Association des Annales de l’institut Fourier},
title = {Conformally bending three-manifolds with boundary},
url = {http://eudml.org/doc/116340},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Gursky, Matthew
AU - Streets, Jeffrey
AU - Warren, Micah
TI - Conformally bending three-manifolds with boundary
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2421
EP - 2447
AB - Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant. This condition roughly means that the curvature is “almost negative”, in a scale-invariant sense.
LA - eng
KW - Almost negative curvature; conformal filling; fully nonlinear equations; manifold with boundary; conformal bending; scalar curvature; pinching of sectional curvature; almost negative curvature
UR - http://eudml.org/doc/116340
ER -

References

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  1. Patricio Aviles, Robert C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds, Duke Math. J. 56 (1988), 395-398 Zbl0645.53023MR932852
  2. C. Bavard, Courbure presque négative en dimension 3 , Compositio Math. 63 (1987), 223-236 Zbl0642.53047MR906372
  3. Peter Buser, Detlef Gromoll, On the almost negatively curved 3 -sphere, Geometry and analysis on manifolds (Katata/Kyoto, 1987) 1339 (1988), 78-85, Springer, Berlin Zbl0651.53032MR961474
  4. Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), 333-363 Zbl0469.35022MR649348
  5. Matthew Gursky, Jeffrey Streets, Micah Warren, Complete conformal metrics of negative Ricci curvature on manifolds with boundary Zbl1214.53035
  6. N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 75-108 Zbl0578.35024MR688919
  7. Charles Loewner, Louis Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers) (1974), 245-272, Academic Press, New York Zbl0298.35018MR358078
  8. Joachim Lohkamp, Negative bending of open manifolds, J. Differential Geom. 40 (1994), 461-474 Zbl0840.53023MR1305978

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