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Evolution of convex entire graphs by curvature flows

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow.
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Wydawca
Czasopismo
Rocznik
Tom
1
Numer
1
Opis fizyczny
Daty
otrzymano
2014-10-21
zaakceptowano
2015-04-20
online
2015-11-03
Twórcy
  • Mathematisches Institut, Albert-Ludwigs-Universität Freiburg
  • Dipartimento di Matematica, Università di Roma “Tor Vergata”
Bibliografia
  • [1] R. Alessandroni, Evolution of hypersurfaces by curvature functions, PhD Thesis, Università di Roma “Tor Vergata”, 2008.
  • [2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2, (1994), 151–171.
  • [3] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217, (1994), 179–197.
  • [4] B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math. 608 (2007),17–33.[WoS]
  • [5] B. Andrews, J. McCoy, Y. Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47(2013), 611–665.[WoS]
  • [6] E. Cabezas Rivas, B. Wilking, How to produce a Ricci Flow via Cheeger-Gromoll exhaustion, J. Eur. Math. Soc., to appear.
  • [7] J. Clutterbuck, O.C. Schnürer, Stability of mean convex cones under mean curvature flow, Math. Z. 267 (2011), 535–547.[WoS]
  • [8] J. Clutterbuck, O.C. Schnürer, F. Schulze, Stability of translating solutions to mean curvature flow, Calc. Var. Partial DifferentialEquations 29 (2007), 281–293.
  • [9] K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. Math. 130 (1989) 453–471.
  • [10] K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991) 547–569.
  • [11] M. Franzen, Existence of convex entire graphs evolving by powers of the mean curvature, arXiv:1112.4359 (2011).
  • [12] R.S. Hamilton, Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2 (1994), 167–172.
  • [13] J. Holland, Interior estimates for hypersurfaces evolving by their k-th Weingarten curvature and some applications, IndianaUniv. Math. J. 63 (2014), 1281–1310.[WoS]
  • [14] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266.
  • [15] G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, (1996).
  • [16] K. Rasul, Slow convergence of graphs under mean curvature flow, Comm. Anal. Geom. 18 (2010), 987–1008.[Crossref]
  • [17] O. Schnürer, J. Urbas, Gauss curvature flows of entire graphs, in preparation. A description of the results can be found onhttp://www.math.uni-konstanz.de/ schnuere/skripte/regensburg.pdf.
  • [18] F. Schulze, M. Simon, Expanding solitons with non-negative curvature operator coming out of cones, MathematischeZeitschrift, 275 (2013) 625–639.[WoS]
  • [19] N. Stavrou, Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math. 499 (1998), 189–198.
  • [20] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_geofl-2015-0006
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