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

Xiaoli Chao; Yusha Lv

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 1, page 195-217
  • ISSN: 0011-4642

Abstract

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We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is δ -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.

How to cite

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Chao, Xiaoli, and Lv, Yusha. "$L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality." Czechoslovak Mathematical Journal 68.1 (2018): 195-217. <http://eudml.org/doc/294484>.

@article{Chao2018,
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.},
author = {Chao, Xiaoli, Lv, Yusha},
journal = {Czechoslovak Mathematical Journal},
keywords = {weighted Poincaré inequality; $\delta $-stability; $L^\{p\}$ harmonic $1$-form; property $(\mathcal \{P\}_\rho )$},
language = {eng},
number = {1},
pages = {195-217},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality},
url = {http://eudml.org/doc/294484},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Chao, Xiaoli
AU - Lv, Yusha
TI - $L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 195
EP - 217
AB - 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.
LA - eng
KW - weighted Poincaré inequality; $\delta $-stability; $L^{p}$ harmonic $1$-form; property $(\mathcal {P}_\rho )$
UR - http://eudml.org/doc/294484
ER -

References

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  1. Calderbank, D. M. J., Gauduchon, P., Herzlich, M., 10.1006/jfan.2000.3563, J. Funct. Anal. 173 (2000), 214-255. (2000) Zbl0960.58010MR1760284DOI10.1006/jfan.2000.3563
  2. Carron, G., 10.1007/s002080050310, Math. Ann. 314 (1999), 613-639 French. (1999) Zbl0933.35054MR1709104DOI10.1007/s002080050310
  3. Cavalcante, M. P., Mirandola, H., Vitório, F., 10.1007/s12220-012-9334-0, J. Geom. Anal. 24 (2014), 205-222. (2014) Zbl1308.53056MR3145922DOI10.1007/s12220-012-9334-0
  4. Chao, X., Lv, Y., 10.4134/JKMS.j150190, J. Korean Math. Soc. 53 (2016), 583-595. (2016) Zbl1339.53059MR3498284DOI10.4134/JKMS.j150190
  5. Dung, N. T., Seo, K., 10.1007/s10455-011-9293-x, Ann. Global Anal. Geom. 41 (2012), 447-460. (2012) Zbl1242.53073MR2891296DOI10.1007/s10455-011-9293-x
  6. Dung, N. T., Seo, K., 10.1016/j.jmaa.2014.10.076, J. Math. Anal. Appl. 423 (2015), 1594-1609. (2015) Zbl1303.53067MR3278217DOI10.1016/j.jmaa.2014.10.076
  7. Fu, H.-P., Li, Z.-Q., 10.2996/kmj/1257948888, Kodai Math. J. 32 (2009), 432-441. (2009) Zbl1182.53048MR2582010DOI10.2996/kmj/1257948888
  8. Greene, R. E., Wu, H., 10.1007/BF01425500, Invent. Math. 27 (1974), 265-298. (1974) Zbl0342.31003MR0382723DOI10.1007/BF01425500
  9. Greene, R. E., Wu, H., 10.1307/mmj/1029002458, Mich. Math. J. 28 (1981), 63-81. (1981) Zbl0477.53058MR0600415DOI10.1307/mmj/1029002458
  10. Hoffman, D., Spruck, J., 10.1002/cpa.3160270601, Commun. Pure Appl. Math. 27 (1974), 715-727. (1974) Zbl0295.53025MR0365424DOI10.1002/cpa.3160270601
  11. Kawai, S., 10.14492/hokmj/1381517802, Hokkaido Math. J. 17 (1988), 147-150. (1988) Zbl0653.53044MR0945852DOI10.14492/hokmj/1381517802
  12. Lam, K.-H., 10.1090/S0002-9947-10-04894-4, Trans. Am. Math. Soc. 362 (2010), 5043-5062. (2010) Zbl1201.53041MR2657671DOI10.1090/S0002-9947-10-04894-4
  13. Li, P., 10.1017/CBO9781139105798, Cambridge Studies in Advanced Mathematics 134, Cambridge University Press, Cambridge (2012). (2012) Zbl1246.53002MR2962229DOI10.1017/CBO9781139105798
  14. Li, P., Schoen, R., 10.1007/BF02392380, Acta Math. 153 (1984), 279-301. (1984) Zbl0556.31005MR0766266DOI10.1007/BF02392380
  15. Li, P., Wang, J., 10.4310/jdg/1090348357, J. Differ. Geom. 58 (2001), 501-534. (2001) Zbl1032.58016MR1906784DOI10.4310/jdg/1090348357
  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. (1993) Zbl0912.53042MR1361194
  17. Palmer, B., 10.1007/BF02566644, Comment. Math. Helv. 66 (1991), 185-188. (1991) Zbl0736.53054MR1107838DOI10.1007/BF02566644
  18. Sang, N. D., Thanh, N. T., 10.4134/CKMS.2014.29.1.123, Commum. Korean. Math. Soc. 29 (2014), 123-130. (2014) Zbl1288.53055MR3162987DOI10.4134/CKMS.2014.29.1.123
  19. Seo, K., 10.1016/j.jmaa.2010.05.048, J. Math. Anal. Appl. 371 (2010), 546-551. (2010) Zbl1195.53087MR2670132DOI10.1016/j.jmaa.2010.05.048
  20. Seo, K., 10.1007/s00013-009-0096-2, Arch. Math. 94 (2010), 173-181. (2010) Zbl1185.53069MR2592764DOI10.1007/s00013-009-0096-2
  21. Seo, K., 10.2140/pjm.2014.268.205, Pac. J. Math. 268 (2014), 205-229. (2014) Zbl1295.53067MR3207607DOI10.2140/pjm.2014.268.205
  22. Shiohama, K., Xu, H., 10.1023/A:1000189116072, Compos. Math. 107 (1997), 221-232. (1997) Zbl0905.53038MR1458750DOI10.1023/A:1000189116072
  23. Tam, L.-F., Zhou, D., 10.1090/S0002-9939-09-09962-6, Proc. Am. Math. Soc. 137 (2009), 3451-3461. (2009) Zbl1184.53016MR2515414DOI10.1090/S0002-9939-09-09962-6
  24. Vieira, M., 10.1007/s10711-016-0165-1, Geom. Dedicata 184 (2016), 175-191. (2016) Zbl1353.53047MR3547788DOI10.1007/s10711-016-0165-1
  25. Yau, S.-T., 10.1512/iumj.1976.25.25051, Indiana Univ. Math. J. 25 (1976), 659-670 erratum ibid. 31 1982 607. (1976) Zbl0335.53041MR0417452DOI10.1512/iumj.1976.25.25051
  26. Yun, G., 10.1023/A:1014211121535, Geom. Dedicata. 89 (2002), 135-141. (2002) Zbl1002.53042MR1890955DOI10.1023/A:1014211121535

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