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)

 
 

 

The self-shrinker in warped product space and the weighted Minkowski inequality


Author: Guoqiang Wu
Journal: Proc. Amer. Math. Soc. 145 (2017), 1763-1772
MSC (2010): Primary 53C44; Secondary 53C24
DOI: https://doi.org/10.1090/proc/13325
Published electronically: October 18, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper consists of two parts. One is that for a kind of self-shrinker in a manifold with warped product metric, we prove that under some conditions on ambient space, the mean convex self-shrinker must have parallel second fundamental form. The other one is a generalization of Brendle's Minkowski inequality for weighted mean curvature.


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

  • [1] Simon Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247-269. MR 3090261, https://doi.org/10.1007/s10240-012-0047-5
  • [2] Simon Brendle and Michael Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 (2013), no. 3, 387-407. MR 3080487
  • [3] Simon Brendle, Pei-Ken Hung, and Mu-Tao Wang, A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold, Comm. Pure Appl. Math. 69 (2016), no. 1, 124-144. MR 3433631, https://doi.org/10.1002/cpa.21556
  • [4] Xu Cheng, Tito Mejia, and Detang Zhou, Eigenvalue estimate and compactness for closed $ f$-minimal surfaces, Pacific J. Math. 271 (2014), no. 2, 347-367. MR 3267533, https://doi.org/10.2140/pjm.2014.271.347
  • [5] Xu Cheng, Tito Mejia, and Detang Zhou, Simons-type equation for $ f$-minimal hypersurfaces and applications, J. Geom. Anal. 25 (2015), no. 4, 2667-2686. MR 3427142, https://doi.org/10.1007/s12220-014-9530-1
  • [6] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497
  • [7] Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153-179. MR 862046
  • [8] Gerhard Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285-299. MR 1030675
  • [9] H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187-197. MR 0238229
  • [10] Jie Wu, A new characterization of geodesic spheres in the hyperbolic space, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3077-3084. MR 3487237, https://doi.org/10.1090/proc/12325
  • [11] Jie Wu and Chao Xia, Hypersurfaces with constant curvature quotients in warped product manifolds, Pacific J. Math. 274 (2015), no. 2, 355-371. MR 3332908, https://doi.org/10.2140/pjm.2015.274.355
  • [12] Jie Wu and Chao Xia, On rigidity of hypersurfaces with constant curvature functions in warped product manifolds, Ann. Global Anal. Geom. 46 (2014), no. 1, 1-22. MR 3205799, https://doi.org/10.1007/s10455-013-9405-x

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C44, 53C24

Retrieve articles in all journals with MSC (2010): 53C44, 53C24


Additional Information

Guoqiang Wu
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200000, People’s Republic of China
Email: gqwu@math.ecnu.edu.cn

DOI: https://doi.org/10.1090/proc/13325
Keywords: Self-shrinker, warped product, Minkowski inequality
Received by editor(s): March 1, 2016
Received by editor(s) in revised form: June 3, 2016
Published electronically: October 18, 2016
Communicated by: Lei Ni
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society