Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Complex Manifolds

Ed. by Fino, Anna Maria

Covered by Web of Science - Emerging Sources Citation Index and Zentralblatt Math (zbMATH)


CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.643
Source Normalized Impact per Paper (SNIP) 2018: 0.812

Mathematical Citation Quotient (MCQ) 2018: 0.61

Open Access
Online
ISSN
2300-7443
See all formats and pricing
More options …

A survey on Inverse mean curvature flow in ROSSes

Giuseppe Pipoli
  • Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universitá degli Studi dell’Aquila, via Vetoio s.n.c., 67100 L’Aquila, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/coma-2017-0016

Abstract

In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.

Keywords : Inverse mean curvature flow; Sub-Riemannian geometry; Webster curvature; Qc curvature

MSC 2010: 53C17; 53C40; 53C44

References

  • [Be] A.L. Besse, Manifolds all of whose geodesics are closed Springer-Verlag, Berlin, Hidelberg, New York, 1978.Google Scholar

  • [Bi] O. Biquard, Métriques d’Einstein asymptotiquement symétriques, Astérisque 265 (2000).Google Scholar

  • [BM] S. Brendle, F.C. Marques, Recent progress on the Yamabe problem, arXiv:1010.4960, (2010).Google Scholar

  • [DT] S. Dragomir, G. Tomassini, Differential geometry and analysis on CR manifolds, Progress in math. vol 246, Birkhäuser (2006).Google Scholar

  • [Di] Q. Ding, The inverse mean curvature flow in rotationally symmetric spaces, Chin. Ann. Math. 32B(1) (2011), 27 - 44.Web of ScienceGoogle Scholar

  • [Du] D. Duchemin Quaternionic contact structure in dimension 7, Ann. Institut Fourier 56(4) (2006), 851 - 885CrossrefGoogle Scholar

  • [Ge1] C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geom, 32 (1990), 299 - 314.Google Scholar

  • [Ge2] C. Gerhardt, Inverse mean curvature flow in hyperbolic space, J. Differential Geom, 89 (2011), 487 - 527.Google Scholar

  • [He] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Accademic press, New York, San Francisco, London, (1978).Google Scholar

  • [HI] G. Huisken, T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom, 59(3) (2001), 353 - 437.Google Scholar

  • [HP] G. Huisken, A. Polden, Geometric evolution equations for hypersurfaces, Hildebrandt S., Struwe M. (eds) Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, vol 1713. Springer, Berlin, Heidelberg (1999), 45 - 84.Google Scholar

  • [Hu] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84(3) (1986), 463 - 480.Google Scholar

  • [HW] P.K. Hung, M.T. Wang, Inverse mean curvature flow in the hyperbolic 3-space revisited, Calculus of variation and partial differential equations 54(1) (2015), 119 - 126.Google Scholar

  • [IMV] S. Ivanov, I. Minchev, D. Vassilev, Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group, arXiv:1504.03142 .Google Scholar

  • [IV1] S. Ivanov, D. Vassilev, Extremal for the Sobolev inequality and the quaternionic contanct Yamabe problem, Imperial College Press Lecture Notes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2011).Google Scholar

  • [IV2] S. Ivanov, D. Vassilev, The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, arXiv:1504.03259 .Google Scholar

  • [JL] D. Jerison, J. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. AMS 1(1), (1988), 1 - 13.Web of ScienceGoogle Scholar

  • [KS] N.Koike, Y. Sakai, The inverse mean curvature flow in rank one symmetric spaces of non-compact type, Kyushu J. Math. 69 (2015), 259 - 284.Web of ScienceGoogle Scholar

  • [Kr] N.V. Krylov, Nonlinear elliptic and parabolic equations of the secondorder, Mathematics and its Applications (Soviet Series), 7, D. Reidel Publishing Co., Dordrecht, (1987).Google Scholar

  • [LP] J.M. Lee, T.H. Parker, The Yamabe problem, Bulletin (new series) of the America mathematical society, 17 (1), (1987), 37 -91.CrossrefGoogle Scholar

  • [Ne] A. Neves, Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds, J. Differential Geom 84 (2016), 191 - 229.Google Scholar

  • [NR] R. Niebergall, P.J. Ryan, Tight and taut Submanifolds, MSRI Pubblications 32, (1997).Google Scholar

  • [Ob] M. Obata, The conjectures on conformai transformations of Riemannian manifolds, J. Differential Geom 6, (1971), 247 -258.Google Scholar

  • [Pa1] J. R. Parker, Notes on complex hyperbolic geometry, http://maths.dur.ac.uk/dma0jrp/img/NCHG.pdf (2003).Google Scholar

  • [Pa2] J. R. Parker, Hyperbolic spaces, http://maths.dur.ac.uk/dma0jrp/img/HSjyvaskyla.pdf (2007).Google Scholar

  • [Pi1] G. Pipoli, Mean curvature flow and Riemannian submersions, Geom. Dedicata 184(1) (2016), 67 - 81.Google Scholar

  • [Pi2] G. Pipoli, Inverse mean curvature flow in complex hyperbolic space, preprint arXiv:1610.01886.Google Scholar

  • [Pi3] G. Pipoli, Inverse mean curvature flow in quaternionic hyperbolic space, to appear on Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.Google Scholar

  • [Sc] J. Scheuer, The inverse mean curvature flow in warped cylinders of non-positive radial curvature, Adv. Math. 306 (2017), 1130 - 1163.Google Scholar

  • [Ur] J.I.E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205, (1990), 355 - 372.Google Scholar

  • [Zh] H. Zhou, Inverse mean curvature flows in warped product manifolds to appear on The Journal of Geometric Analysis.Google Scholar

About the article

Received: 2017-10-15

Accepted: 2017-12-21

Published Online: 2017-12-29

Published in Print: 2017-12-20


Citation Information: Complex Manifolds, Volume 4, Issue 1, Pages 245–262, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2017-0016.

Export Citation

© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in