Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 087, 15 pages      arXiv:2311.02359      https://doi.org/10.3842/SIGMA.2023.087

Deformation of the Weighted Scalar Curvature

Pak Tung Ho a and Jinwoo Shin b
a) Department of Mathematics, Tamkang University, Tamsui, New Taipei City 251301, Taiwan
b) Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea

Received December 07, 2022, in final form October 30, 2023; Published online November 04, 2023

Abstract
Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal $L_\phi^2$-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space, and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.

Key words: weighted scalar curvature; smooth metric measure space; vacuum static space.

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References

  1. Abedin F., Corvino J., On the $P$-scalar curvature, J. Geom. Anal. 27 (2017), 1589-1623.
  2. Andrade M., Cruz T., Silva Santos A., On the $\sigma_2$-curvature and volume of compact manifolds, Ann. Mat. Pura Appl. 202 (2023), 367-395, arXiv:2111.08670.
  3. Bakry D., Émery M., Diffusions hypercontractives, in Séminaire de Probabilités, XIX, 1983/84, Lecture Notes in Math., Vol. 1123, Springer, Berlin, 1985, 177-206.
  4. Berger M., Ebin D., Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379-392.
  5. Case J.S., Smooth metric measure spaces and quasi-Einstein metrics, Internat. J. Math. 23 (2012), 1250110, 36 pages, arXiv:1011.2723.
  6. Case J.S., Smooth metric measure spaces, quasi-Einstein metrics, and tractors, Cent. Eur. J. Math. 10 (2012), 1733-1762, arXiv:1110.3009.
  7. Case J.S., The weighted $\sigma_k$-curvature of a smooth metric measure space, Pacific J. Math. 299 (2019), 339-399, arXiv:1608.01663.
  8. Case J.S., Lin Y.-J., Yuan W., Conformally variational Riemannian invariants, Trans. Amer. Math. Soc. 371 (2019), 8217-8254, arXiv:1711.05579.
  9. Case J.S., Lin Y.-J., Yuan W., Some constructions of formally self-adjoint conformally covariant polydifferential operators, Adv. Math. 401 (2022), 108312, 50 pages, arXiv:2002.05874.
  10. Catino G., Mantegazza C., Mazzieri L., Rimoldi M., Locally conformally flat quasi-Einstein manifolds, J. Reine Angew. Math. 675 (2013), 181-189, arXiv:1010.1418.
  11. Corvino J., Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), 137-189.
  12. Corvino J., Eichmair M., Miao P., Deformation of scalar curvature and volume, Math. Ann. 357 (2013), 551-584, arXiv:1211.6168.
  13. Corvino J., Schoen R.M., On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom. 73 (2006), 185-217, arXiv:gr-qc/0301071.
  14. Cruz T., Vitório F., Prescribing the curvature of Riemannian manifolds with boundary, Calc. Var. Partial Differential Equations 58 (2019), 124, 19 pages, arXiv:1810.01311.
  15. Fischer A.E., Marsden J.E., Deformations of the scalar curvature, Duke Math. J. 42 (1975), 519-547.
  16. He C., Petersen P., Wylie W., On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), 271-311, arXiv:1010.5488.
  17. Ho P.T., Huang Y.-C., Deformation of the scalar curvature and the mean curvature, arXiv:2008.11893.
  18. Kazdan J.L., Warner F.W., Curvature functions for compact $2$-manifolds, Ann. of Math. 99 (1974), 14-47.
  19. Kazdan J.L., Warner F.W., A direct approach to the determination of Gaussian and scalar curvature functions, Invent. Math. 28 (1975), 227-230.
  20. Kazdan J.L., Warner F.W., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. 101 (1975), 317-331.
  21. Kazdan J.L., Warner F.W., Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry 10 (1975), 113-134.
  22. Khaitan A., Weighted GJMS operators on smooth metric measure spaces, arXiv:2203.04719.
  23. Khaitan A., Weighted renormalized volume coefficients, arXiv:2205.06018.
  24. Kim D.-S., Kim Y.H., Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc. 131 (2003), 2573-2576.
  25. Kobayashi O., A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), 665-675.
  26. Lafontaine J., Sur la géométrie d'une généralisation de l'équation différentielle d'Obata, J. Math. Pures Appl. 62 (1983), 63-72.
  27. Lin Y.-J., Yuan W., Deformations of $Q$-curvature I, Calc. Var. Partial Differential Equations 55 (2016), 101, 29 pages, arXiv:1512.05389.
  28. Perelman G., The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
  29. Qing J., Yuan W., A note on static spaces and related problems, J. Geom. Phys. 74 (2013), 18-27, arXiv:1212.1438.
  30. Qing J., Yuan W., On scalar curvature rigidity of vacuum static spaces, Math. Ann. 365 (2016), 1257-1277, arXiv:1412.1860.
  31. Santos A.S., Andrade M., Deformation of the $\sigma_2$-curvature, Ann. Global Anal. Geom. 54 (2018), 71-85, arXiv:1801.00628.

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