Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Nonhomogeneous inverse Gauss curvature flow in $\mathbb {H}^{3}$
HTML articles powered by AMS MathViewer

by Haizhong Li and Tailong Zhou PDF
Proc. Amer. Math. Soc. 147 (2019), 3995-4005 Request permission

Abstract:

In this paper, we consider nonhomogeneous inverse Gauss curvature flows in hyperbolic space $\mathbb {H}^{3}$. We prove that if the initial hypersurface $\Sigma _0$ has nonnegative scalar curvature, then the evolving surface $\Sigma _{t}$ has nonnegative scalar curvature along the flow for $t>0$. The solution of the flow exists for all time and becomes more and more umbilic as $t\rightarrow \infty$.
References
  • Ben Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J. Differential Geom. 39 (1994), no. 2, 407–431. MR 1267897
  • Ben Andrews and Xuzhong Chen, Curvature flow in hyperbolic spaces, J. Reine Angew. Math. 729 (2017), 29–49. MR 3680369, DOI 10.1515/crelle-2014-0121
  • B. Andrews, X. Chen, Y. Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space, arXiv:1805.11776v1, 2018.
  • Ben Andrews and Yong Wei, Quermassintegral preserving curvature flow in hyperbolic space, Geom. Funct. Anal. 28 (2018), no. 5, 1183–1208. MR 3856791, DOI 10.1007/s00039-018-0456-9
  • Bennett Chow and Dan Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004. MR 2061425, DOI 10.1090/surv/110
  • Claus Gerhardt, Curvature problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006. MR 2284727
  • Claus Gerhardt, Inverse curvature flows in hyperbolic space, J. Differential Geom. 89 (2011), no. 3, 487–527. MR 2879249
  • H. Li, X. Wang, Y. Wei, Surfaces expanding by non-concave curvature functions, Ann. Glob. Anal. Geom., arXiv:1609.00570 (2018).
  • Julian Scheuer, Non-scale-invariant inverse curvature flows in hyperbolic space, Calc. Var. Partial Differential Equations 53 (2015), no. 1-2, 91–123. MR 3336314, DOI 10.1007/s00526-014-0742-9
  • Julian Scheuer, Gradient estimates for inverse curvature flows in hyperbolic space, Geom. Flows 1 (2015), no. 1, 11–16. MR 3338988, DOI 10.1515/geofl-2015-0002
  • Y. Wei, New pinching estimates for inverse curvature flows in space forms, J. Geom. Anal., arXiv:1709.02546 (2018).
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C44
  • Retrieve articles in all journals with MSC (2010): 53C44
Additional Information
  • Haizhong Li
  • Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
  • MR Author ID: 255846
  • Email: hli@math.tsinghua.edu.cn
  • Tailong Zhou
  • Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
  • Email: zhou-tl14@mails.tsinghua.edu.cn
  • Received by editor(s): October 10, 2018
  • Received by editor(s) in revised form: January 6, 2019
  • Published electronically: April 18, 2019
  • Additional Notes: This research was supported by NSFC grant No.11671224 and NSFC grant No.11831005
  • Communicated by: Guofang Wei
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 3995-4005
  • MSC (2010): Primary 53C44
  • DOI: https://doi.org/10.1090/proc/14549
  • MathSciNet review: 3993791