Evolution of convex entire graphs by curvature flows

Roberta Alessandroni; Carlo Sinestrari

Geometric Flows (2015)

  • Volume: 1, Issue: 1, page 541-571
  • ISSN: 2353-3382

Abstract

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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.

How to cite

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Roberta Alessandroni, and Carlo Sinestrari. "Evolution of convex entire graphs by curvature flows." Geometric Flows 1.1 (2015): 541-571. <http://eudml.org/doc/276016>.

@article{RobertaAlessandroni2015,
abstract = {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.},
author = {Roberta Alessandroni, Carlo Sinestrari},
journal = {Geometric Flows},
keywords = {initial shape; convex; pinching condition; neckpinch singularity},
language = {eng},
number = {1},
pages = {541-571},
title = {Evolution of convex entire graphs by curvature flows},
url = {http://eudml.org/doc/276016},
volume = {1},
year = {2015},
}

TY - JOUR
AU - Roberta Alessandroni
AU - Carlo Sinestrari
TI - Evolution of convex entire graphs by curvature flows
JO - Geometric Flows
PY - 2015
VL - 1
IS - 1
SP - 541
EP - 571
AB - 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.
LA - eng
KW - initial shape; convex; pinching condition; neckpinch singularity
UR - http://eudml.org/doc/276016
ER -

References

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  1. [1] R. Alessandroni, Evolution of hypersurfaces by curvature functions, PhD Thesis, Università di Roma “Tor Vergata”, 2008. 
  2. [2] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2, (1994), 151– 171. 
  3. [3] B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217, (1994), 179–197. Zbl0807.53044
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  9. [9] K. Ecker, G. Huisken, Mean curvature evolution of entire graphs, Ann. Math. 130 (1989) 453–471. Zbl0696.53036
  10. [10] K. Ecker, G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991) 547–569. Zbl0707.53008
  11. [11] M. Franzen, Existence of convex entire graphs evolving by powers of the mean curvature, arXiv:1112.4359 (2011). 
  12. [12] R.S. Hamilton, Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2 (1994), 167–172. Zbl0843.53002
  13. [13] J. Holland, Interior estimates for hypersurfaces evolving by their k-th Weingarten curvature and some applications, Indiana Univ. Math. J. 63 (2014), 1281–1310. [WoS] Zbl1306.53060
  14. [14] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237–266. Zbl0556.53001
  15. [15] G. M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, (1996). Zbl0884.35001
  16. [16] K. Rasul, Slow convergence of graphs under mean curvature flow, Comm. Anal. Geom. 18 (2010), 987–1008. [Crossref] Zbl1226.53068
  17. [17] O. Schnürer, J. Urbas, Gauss curvature flows of entire graphs, in preparation. A description of the results can be found on http://www.math.uni-konstanz.de/ schnuere/skripte/regensburg.pdf. 
  18. [18] F. Schulze, M. Simon, Expanding solitons with non-negative curvature operator coming out of cones, Mathematische Zeitschrift, 275 (2013) 625–639. [WoS] Zbl1278.53072
  19. [19] N. Stavrou, Selfsimilar solutions to the mean curvature flow, J. Reine Angew. Math. 499 (1998), 189–198. Zbl0895.53039
  20. [20] K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math. 38 (1985), 867–882. Zbl0612.53005

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