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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 |
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.
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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