Non-collapsing for a fully nonlinear inverse curvature flow

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

Non-collapsing for a fully nonlinear inverse curvature flow

Yannan Liu ^1,, and Hongjie Ju ^2,

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.

Keywords: Parabolic equation, fully nonlinear inverse curvature flow, noncollapsing.

Mathematics Subject Classification: 35K45, 53A05.

Citation: Yannan Liu, Hongjie Ju. Non-collapsing for a fully nonlinear inverse curvature flow. Communications on Pure and 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.
[2]	B. Andrews, M. 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.
[3]	B. Andrews and M. Langford, Two-sided non-collapsing curvature flows, preprint. arXiv: 1310.0717.
[4]	B. Andrews, X. L. Han, H. 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.
[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.
[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.
[7]	C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff. Geom., 32 (1990), 299-314.
[8]	M. Grayson, Shortening embedded curves, Ann. of Math., 129 (1989), 71-111. doi: 10.2307/1971486.
[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.
[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.
[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.
[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.
[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.
[14]	J. I. E. Urbas, An expansion of convex hypersurfaces, J. Diff. Geom., 33 (1991), 91-125.
[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.
[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.

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.
[2]	B. Andrews, M. 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.
[3]	B. Andrews and M. Langford, Two-sided non-collapsing curvature flows, preprint. arXiv: 1310.0717.
[4]	B. Andrews, X. L. Han, H. 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.
[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.
[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.
[7]	C. Gerhardt, Flow of Nonconvex Hypersurfaces into Spheres, J. Diff. Geom., 32 (1990), 299-314.
[8]	M. Grayson, Shortening embedded curves, Ann. of Math., 129 (1989), 71-111. doi: 10.2307/1971486.
[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.
[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.
[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.
[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.
[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.
[14]	J. I. E. Urbas, An expansion of convex hypersurfaces, J. Diff. Geom., 33 (1991), 91-125.
[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.
[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.

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