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A weighted renormalized curvature for manifolds with density

Author: Jeffrey S. Case
Journal: Proc. Amer. Math. Soc. 145 (2017), 4031-4040
MSC (2010): Primary 53C21; Secondary 53C25, 58E11
Published electronically: March 23, 2017
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Abstract: We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $ v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated $ \mathcal {W}$-functional.

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  • [1] Thomas P. Branson and A. Rod Gover, Variational status of a class of fully nonlinear curvature prescription problems, Calc. Var. Partial Differential Equations 32 (2008), no. 2, 253-262. MR 2389992,
  • [2] Jeffrey S. Case, A notion of the weighted $ \sigma _k$-curvature for manifolds with density, Adv. Math. 295 (2016), 150-194. MR 3488034,
  • [3] Sun-Yung Alice Chang and Hao Fang, A class of variational functionals in conformal geometry, Int. Math. Res. Not. IMRN 7 (2008), Art. ID rnn008, 16. MR 2428306,
  • [4] Sun-Yung Alice Chang, Hao Fang, and C. Robin Graham, A note on renormalized volume functionals, Differential Geom. Appl. 33 (2014), no. suppl., 246-258. MR 3159960,
  • [5] Xu Cheng and Detang Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces, arXiv:1305.4116, (2099), preprint.
  • [6] C. Robin Graham, Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School ``Geometry and Physics'' (Srní, 1999), 2000, pp. 31-42. MR 1758076
  • [7] C. Robin Graham, Extended obstruction tensors and renormalized volume coefficients, Adv. Math. 220 (2009), no. 6, 1956-1985. MR 2493186,
  • [8] C. Robin Graham and Andreas Juhl, Holographic formula for $ Q$-curvature, Adv. Math. 216 (2007), no. 2, 841-853. MR 2351380,
  • [9] Bin Guo and Haizhong Li, The second variational formula for the functional $ \int v^{(6)}(g)dV_g$, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2911-2925. MR 2801632,
  • [10] Jeff A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), no. 2, 283-316. MR 1738176,
  • [11] William Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1803-1806. MR 2373611,

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Additional Information

Jeffrey S. Case
Affiliation: Department of Mathematics, Pennsylvania State University, 109 McAllister Building, University Park, Pennsylvania 16802

Keywords: Smooth metric measure space, manifold with density, renormalized volume, gradient Ricci soliton, $\mathcal{W}$-functional
Received by editor(s): April 1, 2016
Received by editor(s) in revised form: September 30, 2016
Published electronically: March 23, 2017
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society

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