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2012 | 10 | 5 | 1733-1762
Tytuł artykułu

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

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EN
Abstrakty
EN
We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.
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autor
Bibliografia
  • [1] Alt J., The geometry of conformally Einstein metrics with degenerate Weyl tensor, http://arxiv.org/abs/math/0608598
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  • [8] Branson T., Čap A., Eastwood M., Gover A.R., Prolongations of geometric overdetermined systems, Internat. J. Math., 2006, 17(6), 641–664 http://dx.doi.org/10.1142/S0129167X06003655
  • [9] Branson T., Gover A.R., Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature. Comm. Partial Differential Equations, 2005, 30(10–12), 1611–1669 http://dx.doi.org/10.1080/03605300500299943
  • [10] Cao H.-D., Chen Q., On locally conformally flat gradient steady Ricci solitons, Trans. Amer. Math. Soc., 2012, 364(5), 2377–2391 http://dx.doi.org/10.1090/S0002-9947-2011-05446-2
  • [11] Čap A., Gover A.R., Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc., 2002, 354(4), 1511–1548 http://dx.doi.org/10.1090/S0002-9947-01-02909-9
  • [12] Čap A., Slovák J., Parabolic Geometries I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009
  • [13] Case J.S., Smooth metric measure spaces and quasi-Einstein metrics, preprint available at http://arxiv.org/abs/1011.2723
  • [14] Case J.S., The energy of a smooth metric measure space and applications, preprint available at http://arxiv.org/abs/1011.2728
  • [15] Case J.S., Sharp metric obstructions for quasi-Einstein metrics, preprint available at http://arxiv.org/abs/1110.3010
  • [16] Case J., Shu Y.-J., Wei G., Rigidity of quasi-Einstein metrics, Differential Geom. Appl., 2011, 29(1), 93–100 http://dx.doi.org/10.1016/j.difgeo.2010.11.003
  • [17] Catino G., Generalized quasi-Einstein manifolds with harmonic Weyl tensor, preprint available at http://arxiv.org/abs/1012.5405
  • [18] Catino G., Mantegazza C., Mazzieri L., Rimoldi M., Locally conformally flat quasi-Einstein manifolds, preprint available at http://arxiv.org/abs/1010.1418
  • [19] Chang S.-Y.A., Conformal invariants and partial differential equations, Bull. Amer. Math. Soc., 2005, 42(3), 365–393 http://dx.doi.org/10.1090/S0273-0979-05-01058-X
  • [20] Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 1997, 46(3), 406–480
  • [21] Corvino J., Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., 2000, 214(1), 137–189 http://dx.doi.org/10.1007/PL00005533
  • [22] Gover A.R., Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature, J. Geom. Phys., 2010, 60(2), 182–204 http://dx.doi.org/10.1016/j.geomphys.2009.09.016
  • [23] Gover A.R., Nurowski P., Obstructions to conformally Einstein metrics in n dimensions, J. Geom. Phys., 2006, 56(3), 450–484 http://dx.doi.org/10.1016/j.geomphys.2005.03.001
  • [24] Gover A.R., Peterson L.J., Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus, Comm. Math. Phys., 2003, 235(2), 339–378 http://dx.doi.org/10.1007/s00220-002-0790-4
  • [25] Hammerl M., Invariant prolongation of BGG-operators in conformal geometry, Arch. Math. (Brno), 2008, 44(5), 367–384
  • [26] Hammerl M., Somberg P., Souček V., Šilhan J., On a new normalization for tractor covariant derivatives, preprint available at http://arxiv.org/abs/1003.6090
  • [27] He C., Petersen P., Wylie W., On the classification of warped product Einstein metrics, preprint available at http://arxiv.org/abs/1010.5488
  • [28] He C., Petersen P., Wylie W., The space of virtual solutions to the warped product Einstein equation, preprint available at http://arxiv.org/abs/1110.2456
  • [29] Kim D.-S., Kim Y.H., Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc., 2003, 131(8), 2573–2576 http://dx.doi.org/10.1090/S0002-9939-03-06878-3
  • [30] Kobayashi S., Nomizu K., Foundations of Differential Geometry I, Interscience, New York-London, 1963
  • [31] Miao P., Tam L.-F., On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. Partial Differential Equations, 2009, 36(2), 141–171 http://dx.doi.org/10.1007/s00526-008-0221-2
  • [32] Miao P., Tam L.-F., Einstein and conformally flat critical metrics of the volume functional, Trans. Amer. Math. Soc., 2011, 363(6), 2907–2937 http://dx.doi.org/10.1090/S0002-9947-2011-05195-0
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0091-x
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