Some almost-Schur type inequalities and applications on sub-static manifolds

Fanqi Zeng; Fanqi Zeng

doi:10.3934/era.2022145

Electronic Research Archive

2022, Volume 30, Issue 8: 2860-2870. doi: 10.3934/era.2022145

Previous Article Next Article

Research article

Some almost-Schur type inequalities and applications on sub-static manifolds

Fanqi Zeng ^,

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

Received: 27 June 2021 Revised: 07 April 2022 Accepted: 07 April 2022 Published: 24 May 2022

In this paper, we establish some almost-Schur type inequalities on sub-static manifolds, naturally arising in General Relativity. In particular, our results generalize those in ^[1] of Li-Xia for $ r $-th mean curvatures of closed sub-static hypersurfaces in space forms and $ k $-scalar curvatures for closed locally conformally flat sub-static manifolds. Moreover, our results also generalize those of Cheng ^[2].
- almost-Schur type inequalities,
- sub-static manifolds,
- mean curvature,
- Ricci curvature
Citation: Fanqi Zeng. Some almost-Schur type inequalities and applications on sub-static manifolds[J]. Electronic Research Archive, 2022, 30(8): 2860-2870. doi: 10.3934/era.2022145

Related Papers:

Abstract

In this paper, we establish some almost-Schur type inequalities on sub-static manifolds, naturally arising in General Relativity. In particular, our results generalize those in ^[1] of Li-Xia for $ r $-th mean curvatures of closed sub-static hypersurfaces in space forms and $ k $-scalar curvatures for closed locally conformally flat sub-static manifolds. Moreover, our results also generalize those of Cheng ^[2].

References

[1]	J. Li, C. Xia, An integral formula and its applications on sub-static manifolds, J. Differ. Geom., 113 (2019), 493–518. https://doi.org/10.4310/jdg/1573786972 doi: 10.4310/jdg/1573786972
[2]	X. Cheng, An almost-Schur lemma for symmetric $(2, 0)$ tensors and applications, Pac. J. Math., 267 (2014), 325–340.
[3]	C. De Lellis, P. Topping, Almost-Schur lemma, Calc. Var. Partial Differ. Equ., 43 (2012), 347–354. https://doi.org/10.1007/s00526-011-0413-z doi: 10.1007/s00526-011-0413-z
[4]	B. Chow, P. Lu, L. Ni, Hamilton's Ricci Flow, Lectures in Contemporary Mathematics 3, Science Press, American Mathematical Society, 2006.
[5]	Y. Ge, G. Wang, A new conformal invariant on 3-dimensional manifolds, Adv. Math., 249 (2013), 131–160. https://doi.org/10.1016/j.aim.2013.09.009 doi: 10.1016/j.aim.2013.09.009
[6]	Y. Ge, G. Wang, An almost-Schur theorem on 4-dimensional manifolds, Proc. Amer. Math. Soc., 140 (2012), 1041–1044. https://doi.org/10.1090/S0002-9939-2011-11065-7 doi: 10.1090/S0002-9939-2011-11065-7
[7]	X. Cheng, A generalization of almost-Schur lemma for closed Riemannian manifolds, Ann. Glob. Anal. Geom., 43 (2013), 153–160. https://doi.org/10.1007/s10455-012-9339-8 doi: 10.1007/s10455-012-9339-8
[8]	E. R. Barbosa, A note on the almost-Schur lemma on 4-dimensional Riemannian closed manifold, Proc. Amer. Math. Soc., 140 (2012), 4319–4322. https://doi.org/10.1090/S0002-9939-2012-11255-9 doi: 10.1090/S0002-9939-2012-11255-9
[9]	P. T. Ho, Almost Schur lemma for manifolds with boundary, Differ. Geom. Appl., 32 (2014), 97–112. https://doi.org/10.1016/j.difgeo.2013.11.006 doi: 10.1016/j.difgeo.2013.11.006
[10]	K.-K. Kwong, On an Inequality of Andrews, De Lellis, and Topping, J. Geom. Anal., 25 (2013), 108–121.
[11]	Y. Ge, G. Wang, C. Xia, On problems related to an inequality of De Lellis and Topping, Int. Math. Res. Not., 2013 (2012), 4798–4818. https://doi.org/10.1093/imrn/rns196 doi: 10.1093/imrn/rns196
[12]	D. Perez, On nearly umbilical hypersurfaces, Ph.D thesis, 2011.
[13]	J. Wu, De Lellis-Topping inequalities for smooth metric measure spaces, Geom. Dedicata, 169 (2014), 273–281. https://doi.org/10.1007/s10711-013-9855-0 doi: 10.1007/s10711-013-9855-0
[14]	G. Y. Huang, F. Q. Zeng, De Lellis-Topping type inequalities for $f$-Laplacians, Studia Math., 232 (2016), 189–199. https://doi.org/10.4064/sm8236-4-2016 doi: 10.4064/sm8236-4-2016
[15]	A. Freitas, M. Santos, Some Almost-Schur type inequalities for $k$-Bakry-Emery Ricci tensor, Differ. Geom. Appl., 66 (2019), 82–92. https://doi.org/10.1016/j.difgeo.2019.05.009 doi: 10.1016/j.difgeo.2019.05.009
[16]	S. Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci., 117 (2013), 247–269. https://doi.org/10.1007/s10240-012-0047-5 doi: 10.1007/s10240-012-0047-5
[17]	M.-T. Wang, Y.-K. Wang, X. Zhang, Minkowski formulae and Alexandrov theorems in spacetime, J. Differ. Geom., 105 (2017), 249–290. https://doi.org/10.4310/jdg/1486522815 doi: 10.4310/jdg/1486522815
[18]	S. Borghini, On the characterization of static spacetimes with positive cosmological constant, Ph.D thesis, Scuola Normale Superiore, 2018.
[19]	M. Fogagnolo, A. Pinamonti, New integral estimates in substatic Riemannian manifolds and the Alexandrov theorem, preprint, arXiv: math/2105.04672v1.
[20]	A. Freitas, R. Valos, The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: conservation identities and their applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 38 (2021), 1703–1724.
[21]	R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differ. Geom., 8 (1973), 465–477. https://doi.org/10.4310/jdg/1214431802 doi: 10.4310/jdg/1214431802
[22]	J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101 (2000), 283–316.

Reader Comments

Your name:*

Email:*
© 2022 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)