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May  2017, 16(3): 945-952. doi: 10.3934/cpaa.2017045

Non-collapsing for a fully nonlinear inverse curvature flow

1. 

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China

2. 

School of Science, Beijing University of Posts and Telecommunication, Beijing, 100876, China

* Corresponding author

Received  August 2016 Revised  January 2017 Published  February 2017

Fund Project: The first author is supported by NSFC grant (11201011), NSF of Beijing grant (1132002,1172005). The second author is supported by NSFC grant (11301034,11401527,11471050)

In this paper, we study a fully nonlinear inverse curvature flow in Euclidean space, and prove a non-collapsing property for this flow using maximum principle. Precisely, we show that upon some conditions on speed function, the curvature of the largest touching interior ball is bounded by a multiple of the speed.

Citation: Yannan Liu, Hongjie Ju. Non-collapsing for a fully nonlinear inverse curvature flow. Communications on Pure & Applied Analysis, 2017, 16 (3) : 945-952. doi: 10.3934/cpaa.2017045
References:
[1]

B. Andrews, Non-collapsing in mean-convex mean curvature flow, Geometry and Topology, 16 (2012), 1413-1418.  doi: 10.2140/gt.2012.16.1413.  Google Scholar

[2]

B. AndrewsM. Langford and J. McCoy, Non-collapsing in fully non-linear curvature flows, Ann. I. Poincar′e-AN, 30 (2013), 23-32.  doi: 10.1016/j.anihpc.2012.05.003.  Google Scholar

[3]

B. Andrews and M. Langford, Two-sided non-collapsing curvature flows, preprint. arXiv: 1310.0717.  Google Scholar

[4]

B. AndrewsX. L. HanH. Z. Li and Y. Wei, Non-collapsing for hypersurface flows in the sphere and hyperbolic space, Annali Della Scuola Normal Superiore DI Pisa-Classe DI Science, 14 (2015), 331-338.   Google Scholar

[5]

S. Brendle, Embedded minimal tori in S3 and the Lawson conjecture, Acta. Math., 257 (2015), 462-475.  doi: 10.1007/s11511-013-0101-2.  Google Scholar

[6]

S. Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math., 202 (2015), 217-237.  doi: 10.1007/s00222-014-0570-8.  Google Scholar

[7]

C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff. Geom., 32 (1990), 299-314.   Google Scholar

[8]

M. Grayson, Shortening embedded curves, Ann. of Math., 129 (1989), 71-111.  doi: 10.2307/1971486.  Google Scholar

[9]

R. S. Hamilton, An isoperimetric estimate for the Ricci flow on the two-sphere, Ann. of Math. Stud., 137 (1995), 191-200.  doi: 10.1080/09502389500490321.  Google Scholar

[10]

R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Ann. of Math. Stud., 137 (1995), 201-222.  doi: 10.1016/1053-8127(94)00130-3.  Google Scholar

[11]

G. Huisken, An distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133.  doi: 10.4310/AJM.1998.v2.n1.a2.  Google Scholar

[12]

Y. N. Liu and H. J. Ju, Evolution of convex hypersurfaces by a fully nonlinear flow, Nonlinear Analysis, T.M.A., 130 (2016), 47-58.  doi: 10.1016/j.na.2015.09.014.  Google Scholar

[13]

W. M. Sheng and X. J. Wang, Singularity of profile in the mean curvature flow, Methods Appl. Anal., 16 (2009), 139-155.  doi: 10.4310/MAA.2009.v16.n2.a1.  Google Scholar

[14]

J. I. E. Urbas, An expansion of convex hypersurfaces, J. Diff. Geom., 33 (1991), 91-125.   Google Scholar

[15]

J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205 (1990), 355-372.  doi: 10.1007/BF02571249.  Google Scholar

[16]

B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc., 13 (2000), 665-695.  doi: 10.1090/S0894-0347-00-00338-6.  Google Scholar

show all references

References:
[1]

B. Andrews, Non-collapsing in mean-convex mean curvature flow, Geometry and Topology, 16 (2012), 1413-1418.  doi: 10.2140/gt.2012.16.1413.  Google Scholar

[2]

B. AndrewsM. Langford and J. McCoy, Non-collapsing in fully non-linear curvature flows, Ann. I. Poincar′e-AN, 30 (2013), 23-32.  doi: 10.1016/j.anihpc.2012.05.003.  Google Scholar

[3]

B. Andrews and M. Langford, Two-sided non-collapsing curvature flows, preprint. arXiv: 1310.0717.  Google Scholar

[4]

B. AndrewsX. L. HanH. Z. Li and Y. Wei, Non-collapsing for hypersurface flows in the sphere and hyperbolic space, Annali Della Scuola Normal Superiore DI Pisa-Classe DI Science, 14 (2015), 331-338.   Google Scholar

[5]

S. Brendle, Embedded minimal tori in S3 and the Lawson conjecture, Acta. Math., 257 (2015), 462-475.  doi: 10.1007/s11511-013-0101-2.  Google Scholar

[6]

S. Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math., 202 (2015), 217-237.  doi: 10.1007/s00222-014-0570-8.  Google Scholar

[7]

C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff. Geom., 32 (1990), 299-314.   Google Scholar

[8]

M. Grayson, Shortening embedded curves, Ann. of Math., 129 (1989), 71-111.  doi: 10.2307/1971486.  Google Scholar

[9]

R. S. Hamilton, An isoperimetric estimate for the Ricci flow on the two-sphere, Ann. of Math. Stud., 137 (1995), 191-200.  doi: 10.1080/09502389500490321.  Google Scholar

[10]

R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, Ann. of Math. Stud., 137 (1995), 201-222.  doi: 10.1016/1053-8127(94)00130-3.  Google Scholar

[11]

G. Huisken, An distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133.  doi: 10.4310/AJM.1998.v2.n1.a2.  Google Scholar

[12]

Y. N. Liu and H. J. Ju, Evolution of convex hypersurfaces by a fully nonlinear flow, Nonlinear Analysis, T.M.A., 130 (2016), 47-58.  doi: 10.1016/j.na.2015.09.014.  Google Scholar

[13]

W. M. Sheng and X. J. Wang, Singularity of profile in the mean curvature flow, Methods Appl. Anal., 16 (2009), 139-155.  doi: 10.4310/MAA.2009.v16.n2.a1.  Google Scholar

[14]

J. I. E. Urbas, An expansion of convex hypersurfaces, J. Diff. Geom., 33 (1991), 91-125.   Google Scholar

[15]

J. I. E. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., 205 (1990), 355-372.  doi: 10.1007/BF02571249.  Google Scholar

[16]

B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc., 13 (2000), 665-695.  doi: 10.1090/S0894-0347-00-00338-6.  Google Scholar

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