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Keywords:
weighted Poincaré inequality; $\delta $-stability; $L^{p}$ harmonic $1$-form; property $(\mathcal {P}_\rho )$
Summary:
We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $\delta $-stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^p$ harmonic $1$-forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.
References:
[1] Calderbank, D. M. J., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173 (2000), 214-255. DOI 10.1006/jfan.2000.3563 | MR 1760284 | Zbl 0960.58010
[2] Carron, G.: $L^2$-cohomologie et inégalités de Sobolev. Math. Ann. 314 (1999), 613-639 French. DOI 10.1007/s002080050310 | MR 1709104 | Zbl 0933.35054
[3] Cavalcante, M. P., Mirandola, H., Vitório, F.: $L^2$ harmonic $1$-forms on submanifolds with finite total curvature. J. Geom. Anal. 24 (2014), 205-222. DOI 10.1007/s12220-012-9334-0 | MR 3145922 | Zbl 1308.53056
[4] Chao, X., Lv, Y.: $L^2$ harmonic $1$-forms on submanifolds with weighted Poincaré inequality. J. Korean Math. Soc. 53 (2016), 583-595. DOI 10.4134/JKMS.j150190 | MR 3498284 | Zbl 1339.53059
[5] Dung, N. T., Seo, K.: Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature. Ann. Global Anal. Geom. 41 (2012), 447-460. DOI 10.1007/s10455-011-9293-x | MR 2891296 | Zbl 1242.53073
[6] Dung, N. T., Seo, K.: Vanishing theorems for $L^2$ harmonic $1$-forms on complete submanifolds in a Riemannian manifold. J. Math. Anal. Appl. 423 (2015), 1594-1609. DOI 10.1016/j.jmaa.2014.10.076 | MR 3278217 | Zbl 1303.53067
[7] Fu, H.-P., Li, Z.-Q.: $L^2$ harmonic $1$-forms on complete submanifolds in Euclidean space. Kodai Math. J. 32 (2009), 432-441. DOI 10.2996/kmj/1257948888 | MR 2582010 | Zbl 1182.53048
[8] Greene, R. E., Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Invent. Math. 27 (1974), 265-298. DOI 10.1007/BF01425500 | MR 0382723 | Zbl 0342.31003
[9] Greene, R. E., Wu, H.: Harmonic forms on noncompact Riemannian and Kähler manifolds. Mich. Math. J. 28 (1981), 63-81. DOI 10.1307/mmj/1029002458 | MR 0600415 | Zbl 0477.53058
[10] Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27 (1974), 715-727. DOI 10.1002/cpa.3160270601 | MR 0365424 | Zbl 0295.53025
[11] Kawai, S.: Operator $\triangle-aK$ on surfaces. Hokkaido Math. J. 17 (1988), 147-150. DOI 10.14492/hokmj/1381517802 | MR 0945852 | Zbl 0653.53044
[12] Lam, K.-H.: Results on a weighted Poincaré inequality of complete manifolds. Trans. Am. Math. Soc. 362 (2010), 5043-5062. DOI 10.1090/S0002-9947-10-04894-4 | MR 2657671 | Zbl 1201.53041
[13] Li, P.: Geometric Analysis. Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, Cambridge (2012). DOI 10.1017/CBO9781139105798 | MR 2962229 | Zbl 1246.53002
[14] Li, P., Schoen, R.: $L^p$ and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153 (1984), 279-301. DOI 10.1007/BF02392380 | MR 0766266 | Zbl 0556.31005
[15] Li, P., Wang, J.: Complete manifolds with positive spectrum. J. Differ. Geom. 58 (2001), 501-534. DOI 10.4310/jdg/1090348357 | MR 1906784 | Zbl 1032.58016
[16] Miyaoka, R.: $L^2$ harmonic $1$-forms on a complete stable minimal hypersurface. Geometry and Global Analysis T. Kotake et al. Int. Research Inst., Sendai 1993, Tôhoku Univ., Mathematical Institute (1993), 289-293. MR 1361194 | Zbl 0912.53042
[17] Palmer, B.: Stability of minimal hypersurfaces. Comment. Math. Helv. 66 (1991), 185-188. DOI 10.1007/BF02566644 | MR 1107838 | Zbl 0736.53054
[18] Sang, N. D., Thanh, N. T.: Stable minimal hypersurfaces with weighted Poincaré inequality in a Riemannian manifold. Commum. Korean. Math. Soc. 29 (2014), 123-130. DOI 10.4134/CKMS.2014.29.1.123 | MR 3162987 | Zbl 1288.53055
[19] Seo, K.: $L^2$ harmonic $1$-forms on minimal submanifolds in hyperbolic space. J. Math. Anal. Appl. 371 (2010), 546-551. DOI 10.1016/j.jmaa.2010.05.048 | MR 2670132 | Zbl 1195.53087
[20] Seo, K.: Rigidity of minimal submanifolds in hyperbolic space. Arch. Math. 94 (2010), 173-181. DOI 10.1007/s00013-009-0096-2 | MR 2592764 | Zbl 1185.53069
[21] Seo, K.: $L^p$ harmonic $1$-forms and first eigenvalue of a stable minimal hypersurface. Pac. J. Math. 268 (2014), 205-229. DOI 10.2140/pjm.2014.268.205 | MR 3207607 | Zbl 1295.53067
[22] Shiohama, K., Xu, H.: The topological sphere theorem for complete submanifolds. Compos. Math. 107 (1997), 221-232. DOI 10.1023/A:1000189116072 | MR 1458750 | Zbl 0905.53038
[23] Tam, L.-F., Zhou, D.: Stability properties for the higher dimensional catenoid in $\mathbb{R}^{n+1}$. Proc. Am. Math. Soc. 137 (2009), 3451-3461. DOI 10.1090/S0002-9939-09-09962-6 | MR 2515414 | Zbl 1184.53016
[24] Vieira, M.: Vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds. Geom. Dedicata 184 (2016), 175-191. DOI 10.1007/s10711-016-0165-1 | MR 3547788 | Zbl 1353.53047
[25] Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25 (1976), 659-670 erratum ibid. 31 1982 607. DOI 10.1512/iumj.1976.25.25051 | MR 0417452 | Zbl 0335.53041
[26] Yun, G.: Total scalar curvature and $L^2$ harmonic $1$-forms on a minimal hypersurface in Euclidean space. Geom. Dedicata. 89 (2002), 135-141. DOI 10.1023/A:1014211121535 | MR 1890955 | Zbl 1002.53042
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