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AIMS Mathematics

2017, Volume 2, Issue 3: 545-556. doi: 10.3934/Math.2017.2.545
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Research article

Applications of the lichnerowicz Laplacian to stress energy tensors

  • Department of Mathematics, University of Texas, TX 78539-2999 Edinburg, USA
  • Received: 23 June 2017 Accepted: 30 August 2017 Published: 13 September 2017

  • A generalization of the Laplacian for p-forms to arbitrary tensors due to Lichnerowicz will be applied to a 2-tensor which has physical applications. It is natural to associate a divergencefree symmetric 2-tensor to a critical point of a specific variational problem and it is this 2-tensor that is studied. Numerous results are obtained for the stress-energy tensor, such as its divergence and Laplacian. A remarkable integral formula involving a symmetric 2-tensor and a conformal vector field is obtained as well.

    Citation: Paul Bracken. Applications of the lichnerowicz Laplacian to stress energy tensors[J]. AIMS Mathematics, 2017, 2(3): 545-556. doi: 10.3934/Math.2017.2.545

    Related Papers:

  • A generalization of the Laplacian for p-forms to arbitrary tensors due to Lichnerowicz will be applied to a 2-tensor which has physical applications. It is natural to associate a divergencefree symmetric 2-tensor to a critical point of a specific variational problem and it is this 2-tensor that is studied. Numerous results are obtained for the stress-energy tensor, such as its divergence and Laplacian. A remarkable integral formula involving a symmetric 2-tensor and a conformal vector field is obtained as well.


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    [1] P. Baird, Stress-energy tensors and the Lichnerowicz Laplacian, J. of Geometry and Physics, 58(2008), 1329-1342.
    [2] P. Blanchard and E. Brüning, Variational Methods in Mathematical Physics, Springer-Verlag, Berlin-Heidelberg, 1992.
    [3] S. S. Chern, Minimal surfaces in Euclidean Space of N dimensions: Symposium in Honor of Marston Morse, Princeton Univ. Press, 1965,187-198.
    [4] J Eells and L. Lemaine, Selected Topics in Harmonic Maps: CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 50 (1983).
    [5] A. Lichnerowicz, Géometrie des Groups de Transformations, Dunod, Paris, 1958.
    [6] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Publ. Math. Inst. Hautes Ëtudes Sci., 10 (1961), 293-344.
    [7] P. Peterson, Riemannian Geometry, Springer-Verlag, NY, 1998.
    [8] R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure and Applied Math. XLI, 317-392,1988.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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