Abstract
We study space-like self-shrinkers of dimension n in pseudo-Euclidean space R m+n m with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Calabi, E.: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Global Analysis U. C. Berkeley, 1968
Chau, A., Chen, J., Yuan, Y.: Rigidity of Entire self-shrinking solutions to curvature flows. J. Reine Angew. Math., 664, 229–239 (2012)
Chen, Q., Qiu, H. B.: Rigidity theorems for self-shrinker in Euclidean space and Pseudo-Eclideanspace, preprint
Cheng, S. Y., Yau, S. T.: Maximal spacelike hypersurfaces in the Lorentz–Minkowski spaces. Ann. Math., 104, 407–419 (1976)
Cheng, X., Zhou, D. T.: Volume estimates about shrinkers, arXiv:1106.4950
Colding, T. H., Minicozzi, W. P.: Generic mean curvature flow I; generic singularities. Ann. Math., 175, 755–833 (2012)
Ding, Q., Wang, Z. Z.: On the self-shrinking system in arbitrary codimensional spaces, arXiv:1012.0429v2 [math.DG]
Ding, Q., Xin, Y. L., Volume growth, eigenvalue and compactness for self-shrinkers. Asian J. Math., 17, 443–456 (2013)
Ding, Q., Xin, Y. L.: The rigidity theorems for Lagrangian self-shrinkers. J. Reine Angew. Math., accepted, arXiv:1112.2453 [math.DG]
Ecker, K.: On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime. J. Austral. Math. Soc. Ser. A, 55(1), 41–59 (1993)
Ecker, K.: Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J. Differential Geom., 46(3), 481–498 (1997)
Ecker, K.: Mean curvature flow of of spacelike hypersurfaces near null initial data. Comm. Anal. Geom., 11(2), 181–205 (2003)
Ecker, K., Huisken, G.: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys., 135, 595–613 (1991)
Huang, R. L., Wang, Z. Z.: On the entire self-shrinking solutions to Lagrangian mean curvature flow. Calc. Var. Partial Differential Equations, 41, 321–339 (2011)
Halldorsson, H. P.: Self-similar sulutions to the mean curvature flow in the Minkowski plane R1,1, arXiv:1212.0276v1[math.DG]
Jost, J., Xin, Y. L.: Some aspects ofthe global geometry of entire space-like submanifolds. Result Math., 40, 233–245 (2001)
Wong, Y.-C.: Euclidean n-planes in pseudo-Euclidean spaces and differential geometry of Cartan domain. Bull. A. M. S., 75, 409–414 (1969)
Wei, G. F., Wylie, W.: Comparison geometry for Bakry–Emery Ricci tensor. J. Differential Geometry, 83(2), 377–405 (2009)
Xin, Y. L.: Mean curvature flow with bounded Gauss image. Results Math., 59, 415–436 (2011)
Xin, Y. L.: On the Gauss image of a spacelike hypersurfaces with constant mean curvature in Minkowski space. Comment. Math. Helv., 66, 590–598 (1991)
Xin, Y. L.: Minimal Submanifolds and Related Topics, World Scientific Publ., Singapore, 2003
Xin, Y. L., Ye, R. G.: Bernstein-type theorems for space-like surfaces with parallel mean curvature. J. Rein Angew. Math., 489, 189–198 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grant No. 11271072) and He’nan University Seed Fund
Rights and permissions
About this article
Cite this article
Liu, H.Q., Xin, Y.L. Some results on space-like self-shrinkers. Acta. Math. Sin.-English Ser. 32, 69–82 (2016). https://doi.org/10.1007/s10114-014-4082-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-014-4082-7