Some De Lellis-Topping type inequalities and their applications on an NCC Riemannian triple with boundary. (English) Zbl 1515.53038
Summary: In this paper, we prove some De Lellis-Topping type inequalities for an NCC Riemannian triple with null infinitesimally convex boundary. As applications we derive some sharp geometric inequalities for free boundary hypersurfaces in a ball in space forms.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C24 | Rigidity results |
Keywords:
De Lellis-Topping inequalities; Riemannian triples; free boundary hypersurfaces; Reilly formulaReferences:
| [1] | Agostiniani, V.; Mazzieri, L.; Oronzio, F., A geometric capacitary inequality for sub-static manifolds with harmonic potentials, Math. Eng., 4, 40 (2022) · Zbl 1504.53047 |
| [2] | Barbosa, ER, A note on the almost-Schur lemma on 4-dimensional Riemannian closed manifold, Proc. Am. Math. Soc., 140, 4319-4322 (2012) · Zbl 1273.53042 · doi:10.1090/S0002-9939-2012-11255-9 |
| [3] | Bartolo, R.; Germinario, A.; Sánchez, M., A note on the boundary of a static Lorentzian manifold, Differ. Geom. Appl., 16, 121-131 (2002) · Zbl 1036.53047 · doi:10.1016/S0926-2245(02)00062-1 |
| [4] | Caponio, E.; Sánchez, M., Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold, “Recent Trends in Lorentzian Geometry”. Springer Proceedings in Mathematics and Statistics (2013), New York: Springer Science + Business Media, New York |
| [5] | Cheng, X., A generalization of almost-Schur lemma for closed Riemannian manifolds, Ann. Glob. Anal. Geom., 43, 153-160 (2013) · Zbl 1276.53054 · doi:10.1007/s10455-012-9339-8 |
| [6] | Cheng, X.; Zhou, D., Rigidity for closed totally umbilical hypersurfaces in space forms, J. Geom. Anal., 24, 1337-1345 (2014) · Zbl 1303.53064 · doi:10.1007/s12220-012-9375-4 |
| [7] | Cheng, X., An almost-Schur lemma for symmetric \((2, 0)\) tensors and applications, Pac. J. Math., 267, 325-340 (2014) · Zbl 1295.53026 · doi:10.2140/pjm.2014.267.325 |
| [8] | Chen, Y.; Hu, Y.; Li, H., Geometric inequalities for free boundary hypersurfaces in a ball, Ann. Glob. Anal. Geom., 62, 33-45 (2022) · Zbl 1498.53053 · doi:10.1007/s10455-022-09836-2 |
| [9] | Chow, B.; Lu, P.; Ni, L., Hamilton’s Ricci Flow, Lectures in Contemporary Mathematics (2006), Providence: Science Press, American Mathematical Society, Providence · Zbl 1118.53001 |
| [10] | Cruz, F. Jr; Freitas, A.; Santos, M., De Lellis-Topping inequalities on weighted manifolds with boundary, Results Math., 77, 14 (2022) · Zbl 1491.53048 · doi:10.1007/s00025-021-01549-5 |
| [11] | De Lellis, C.; Topping, P., Almost-Schur lemma, Calc. Var. Partial Differ. Equ., 43, 347-354 (2012) · Zbl 1236.53036 · doi:10.1007/s00526-011-0413-z |
| [12] | Freitas, A.; Santos, M., Some almost-Schur type inequalities for \(k\)-Bakry-Emery Ricci tensor, Differ. Geom. Appl., 66, 82-92 (2019) · Zbl 1423.53046 · doi:10.1016/j.difgeo.2019.05.009 |
| [13] | Freitas, A.; Valos, R., The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: conservation identities and their applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 38, 1703-1724 (2021) · Zbl 1483.53058 · doi:10.1016/j.anihpc.2021.01.002 |
| [14] | Fogagnolo, M.; Pinamonti, A., New integral estimates in substatic Riemannian manifolds and the Alexandrov theorem, J. Math. Pures Appl., 163, 299-317 (2022) · Zbl 1496.53056 · doi:10.1016/j.matpur.2022.05.007 |
| [15] | Gilbarg, D.; Trudinger, NS, Elliptic Partial Differential Equations of Second Order (2015), Berlin: Springer, Berlin |
| [16] | Ge, Y.; Wang, G., A new conformal invariant on 3-dimensional manifolds, Adv. Math., 249, 131-160 (2013) · Zbl 1293.53043 · doi:10.1016/j.aim.2013.09.009 |
| [17] | Ge, Y.; Wang, G., An almost-Schur theorem on 4-dimensional manifolds, Proc. Am. Math. Soc., 140, 1041-1044 (2012) · Zbl 1238.53026 · doi:10.1090/S0002-9939-2011-11065-7 |
| [18] | Ge, Y.; Wang, G.; Xia, C., On problems related to an inequality of De Lellis and Topping, Int. Math. Res. Not., 2013, 4798-4818 (2012) · Zbl 1296.53094 · doi:10.1093/imrn/rns196 |
| [19] | Girão, F.; Rodrigues, D., Weighted geometric inequalities for hypersurfaces in sub-static manifolds, Bull. Lond. Math. Soc., 52, 121-136 (2020) · Zbl 1443.52005 · doi:10.1112/blms.12312 |
| [20] | Huang, G.; Zeng, F., De Lellis-Topping type inequalities for \(f\)-Laplacians, Studia Math., 232, 189-199 (2016) · Zbl 1354.53051 |
| [21] | Huang, G., Ma, B., Zhu, M.: A Reilly type integral formula and its applications. arXiv:2201.09439 |
| [22] | Ho, PT, Almost Schur lemma for manifolds with boundary, Differ. Geom. Appl., 32, 97-112 (2014) · Zbl 1283.53038 · doi:10.1016/j.difgeo.2013.11.006 |
| [23] | Kwong, K-K, On an inequality of Andrews, De Lellis, and Topping, J. Geom. Anal., 25, 108-121 (2013) · Zbl 1316.53070 · doi:10.1007/s12220-013-9415-8 |
| [24] | Li, H.; Xiong, C., Stability of capillary hypersurfaces in a Euclidean ball, Pac. J. Math., 297, 131-146 (2018) · Zbl 1401.49063 · doi:10.2140/pjm.2018.297.131 |
| [25] | Li, J.; Xia, C., An integral formula and its applications on sub-static manifolds, J. Differ. Geom., 113, 493-518 (2019) · Zbl 1433.53059 · doi:10.4310/jdg/1573786972 |
| [26] | Meng, M.; Zhang, S., De Lellis-Topping type inequalities on smooth metric measure spaces, Front. Math. China, 13, 147-160 (2018) · Zbl 1390.53030 · doi:10.1007/s11464-017-0670-z |
| [27] | Perez, D.: On nearly umbilical hypersurfaces. Thesis (2011) |
| [28] | Reilly, RC, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differ. Geom., 8, 465-477 (1973) · Zbl 0277.53030 · doi:10.4310/jdg/1214431802 |
| [29] | Wu, J., De Lellis-Topping inequalities for smooth metric measure spaces, Geom. Dedicata, 169, 273-281 (2014) · Zbl 1293.53050 · doi:10.1007/s10711-013-9855-0 |
| [30] | Wang, G.; Xia, C., Uniqueness of stable capillary hypersurfaces in a ball, Math. Ann., 374, 1845-1882 (2019) · Zbl 1444.53040 · doi:10.1007/s00208-019-01845-0 |
| [31] | Wang, M-T; Wang, Y-K; Zhang, X., Minkowski formulae and Alexandrov theorems in spacetime, J. Differ. Geom., 105, 249-290 (2017) · Zbl 1380.53089 · doi:10.4310/jdg/1486522815 |
| [32] | Zeng, F., Some almost-Schur type inequalities and applications on sub-static manifolds, Electron. Res. Arch., 30, 2860-2870 (2022) · Zbl 1539.53073 · doi:10.3934/era.2022145 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
