This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.
Chunlei HE
,
Shoujun HUANG
,
Xiaomin XING
. SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW[J]. Acta mathematica scientia, Series B, 2017
, 37(3)
: 657
-667
.
DOI: 10.1016/S0252-9602(17)30028-0
[1] Chou K S, Wo W F. On hyperbolic Gauss curvature flows. J Differential Geom, 2011, 89:455-485
[2] Gage M, Hamilton R. The heat equation shrinking concex plane curves. J Differential Geom, 1986, 23:417-491
[3] Halldorsson H P. Self-similar solutions to the curve shortening flow. Transactions of the American Mathematical Society, 2012, 364:5285-5309
[4] He C L. Kong D X, Liu K F. Hyperbolic mean curvature flow. J Differential Equations, 2009, 246:373-390
[5] Huisken G, Ilmanen T. The inverse mean curvature flow and the Riemann Penrose inequality. J Differential Geom, 2001, 59:353-437
[6] Kong D X, Liu K F, Wang Z G. Hyperbolic mean curvature flow:evolution of plane curves. Acta Mathematica Scientia, 2009, 29:493-514
[7] Kong D X, Wang Z G. Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow. J Differential Equations, 2009, 247:1694-1719
[8] LeFloch P G, Smoczyk K. The hyperbolic mean curvature flow. J Math Pures Appl, 2008, 90:591-614
[9] Wang Z G. Hyperbolic mean curvature flow in Minkowski space. Nonlinear Analysis, 2014, 94:259-271
[10] Wang Z G. Blow-up of periodic solutions to reducible quasilinear hyperbolic systems. Nonlinear Analysis, 2010, 73:704-712
