Czechoslovak Mathematical Journal, first online, pp. 1-23


$L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality

Xiaoli Chao, Yusha Lv

Received August 4, 2016.   First published January 19, 2018.

Abstract:  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.
Keywords:  weighted Poincaré inequality; $\delta$-stability; $L^p$ harmonic $1$-form; property $(\mathcal{P}_\rho)$
Classification MSC:  53C42, 53C50
DOI:  10.21136/CMJ.2018.0415-16

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] D. M. J. Calderbank, P. Gauduchon, M. Herzlich: 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] G. Carron: $L^2$-cohomologie et inégalités de Sobolev. Math. Ann. 314 (1999), 613-639. (In French.) DOI 10.1007/s002080050310 | MR 1709104 | Zbl 0933.35054
[3] M. P. Cavalcante, H. Mirandola, F. Vitório: $L^2$ 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] X. Chao, Y. Lv: $L^2$ 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] N. T. Dung, K. Seo: 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] N. T. Dung, K. Seo: 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] H.-P. Fu, Z.-Q. Li: $L^2$ 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] R. E. Greene, H. Wu: 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] R. E. Greene, H. Wu: 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] D. Hoffman, J. Spruck: 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] S. Kawai: Operator $\triangle-aK$ on surfaces. Hokkaido Math. J. 17 (1988), 147-150. DOI 10.14492/hokmj/1381517802 | MR 0945852 | Zbl 0653.53044
[12] K.-H. Lam: 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] P. Li: Geometric Analysis. Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, Cambridge (2012). DOI 10.1017/CBO9781139105798 | MR 2962229 | Zbl 1246.53002
[14] P. Li, R. Schoen: $L^p$ 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] P. Li, J. Wang: Complete manifolds with positive spectrum. J. Differ. Geom. 58 (2001), 501-534. DOI 10.4310/jdg/1090348357 | MR 1906784 | Zbl 1032.58016
[16] R. Miyaoka: $L^2$ harmonic $1$-forms on a complete stable minimal hypersurface. Geometry and Global Analysis (T. Kotake et al., eds.). Int. Research Inst., Sendai 1993, Tôhoku Univ., Mathematical Institute (1993), 289-293. MR 1361194 | Zbl 0912.53042
[17] B. Palmer: Stability of minimal hypersurfaces. Comment. Math. Helv. 66 (1991), 185-188. DOI 10.1007/BF02566644 | MR 1107838 | Zbl 0736.53054
[18] N. D. Sang, N. T. Thanh: 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] K. Seo: $L^2$ 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] K. Seo: 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] K. Seo: $L^p$ 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] K. Shiohama, H. Xu: The topological sphere theorem for complete submanifolds. Compos. Math. 107 (1997), 221-232. DOI 10.1023/A:1000189116072 | MR 1458750 | Zbl 0905.53038
[23] L.-F. Tam, D. Zhou: 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] M. Vieira: 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] S.-T. Yau: 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] G. Yun: 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

Affiliations:   Xiaoli Chao, School of Mathematics, Southeast University, 2 Sipailou, Xuanwu, Nanjing 211189, Jiangsu, P. R. China, e-mail: xlchao@seu.edu.cn, Yusha Lv, School of Mathematics, Wuhan University, Meiyuan 2nd Road, Wuhan 430072, Hubei, P. R. China, e-mail: lvyushasx@163.com


 
PDF available at: