Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Rigidity properties of smooth metric measure spaces via the weighted $ p$-Laplacian


Author: Nguyen Thac Dung
Journal: Proc. Amer. Math. Soc. 145 (2017), 1287-1299
MSC (2010): Primary 53C23, 53C24, 58J05
DOI: https://doi.org/10.1090/proc/13285
Published electronically: September 8, 2016
MathSciNet review: 3589326
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show sharp estimates for the first eigenvalue $ \lambda _{1, p}$ of the weighted $ p$-Laplacian on smooth metric measure spaces $ (M, g, e^{-f}dv)$. When the Bakry-Émery curvature $ Ric_f$ is bounded from below and the
weighted function $ f$ is of sublinear growth, we prove some rigidity properties provided that the first eigenvalue $ \lambda _{1, p}$ obtains its optimal value.


References [Enhancements On Off] (What's this?)

  • [1] Walter Allegretto and Yin Xi Huang, A Picone's identity for the $ p$-Laplacian and applications, Nonlinear Anal. 32 (1998), no. 7, 819-830. MR 1618334, https://doi.org/10.1016/S0362-546X(97)00530-0
  • [2] Stephen M. Buckley and Pekka Koskela, Ends of metric measure spaces and Sobolev inequalities, Math. Z. 252 (2006), no. 2, 275-285. MR 2207797, https://doi.org/10.1007/s00209-005-0846-1
  • [3] Shu-Cheng Chang, Jui-Tang Chen, and Shihshu Walter Wei, Liouville properties for $ p$-harmonic maps with finite $ q$-energy, Trans. Amer. Math. Soc. 368 (2016), no. 2, 787-825. MR 3430350, https://doi.org/10.1090/tran/6351
  • [4] B. Kotschwar and L. Ni, Gradient estimate for $ p$-harmonic function, $ 1/H$ flow and an entropy formula, Ann. Sci. Éc. Norm. Supér., 42 (2009), 1-36
  • [5] Peter Li and Jiaping Wang, Complete manifolds with positive spectrum. II, J. Differential Geom. 62 (2002), no. 1, 143-162. MR 1987380
  • [6] Peter Li and Jiaping Wang, Connectedness at infinity of complete Kähler manifolds, Amer. J. Math. 131 (2009), no. 3, 771-817. MR 2530854, https://doi.org/10.1353/ajm.0.0051
  • [7] Peter Li and Luen-Fai Tam, Linear growth harmonic functions on a complete manifold, J. Differential Geom. 29 (1989), no. 2, 421-425. MR 982183
  • [8] Roger Moser, The inverse mean curvature flow and $ p$-harmonic functions, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 1, 77-83. MR 2283104, https://doi.org/10.4171/JEMS/73
  • [9] Ovidiu Munteanu and Jiaping Wang, Smooth metric measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011), no. 3, 451-486. MR 2843238, https://doi.org/10.4310/CAG.2011.v19.n3.a1
  • [10] Ovidiu Munteanu and Jiaping Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55-94. MR 2903101, https://doi.org/10.4310/CAG.2012.v20.n1.a3
  • [11] Nobumitsu Nakauchi, A Liouville type theorem for $ p$-harmonic maps, Osaka J. Math. 35 (1998), no. 2, 303-312. MR 1648911
  • [12] Peter Petersen and William Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2009), no. 2, 329-345. MR 2507581, https://doi.org/10.2140/pjm.2009.241.329
  • [13] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Constancy of $ p$-harmonic maps of finite $ q$-energy into non-positively curved manifolds, Math. Z. 258 (2008), no. 2, 347-362. MR 2357641, https://doi.org/10.1007/s00209-007-0175-7
  • [14] Chiung-Jue Anna Sung and Jiaping Wang, Sharp gradient estimate and spectral rigidity for $ p$-Laplacian, Math. Res. Lett. 21 (2014), no. 4, 885-904. MR 3275651, https://doi.org/10.4310/MRL.2014.v21.n4.a14
  • [15] Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150. MR 727034, https://doi.org/10.1016/0022-0396(84)90105-0
  • [16] Lin Feng Wang, Eigenvalue estimate for the weighted $ p$-Laplacian, Ann. Mat. Pura Appl. (4) 191 (2012), no. 3, 539-550. MR 2958348, https://doi.org/10.1007/s10231-011-0195-0
  • [17] J. Y. Chen and Y. Wang, Liouville type theorems for the p-harmonic functions on certain manifolds, arXiv:1411.1492 (To appear in Pacific Jour. Math.)
  • [18] Xiaodong Wang and Lei Zhang, Local gradient estimate for $ p$-harmonic functions on Riemannian manifolds, Comm. Anal. Geom. 19 (2011), no. 4, 759-771. MR 2880214, https://doi.org/10.4310/CAG.2011.v19.n4.a4
  • [19] Yu-Zhao Wang and Huai-Qian Li, Lower bound estimates for the first eigenvalue of the weighted $ p$-Laplacian on smooth metric measure spaces, Differential Geom. Appl. 45 (2016), 23-42. MR 3457386, https://doi.org/10.1016/j.difgeo.2015.11.008

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C23, 53C24, 58J05

Retrieve articles in all journals with MSC (2010): 53C23, 53C24, 58J05


Additional Information

Nguyen Thac Dung
Affiliation: Department of Mathematics–Mechanics–Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam
Email: dungmath@yahoo.co.uk, dungmath@gmail.com

DOI: https://doi.org/10.1090/proc/13285
Keywords: Weighted $p$-harmonic functions, smooth metric measure spaces, the first eigenvalue, splitting theorems, gradient Ricci solitons
Received by editor(s): March 14, 2016
Received by editor(s) in revised form: April 23, 2016, and May 4, 2016
Published electronically: September 8, 2016
Communicated by: Guofang Wei
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society