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Forced convex mean curvature flow in Euclidean spaces. (English) Zbl 1149.53039
Summary: We show that the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exists for all times and expands to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35K55 Nonlinear parabolic equations 53A07 Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
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##### References:
 [1] Aarons M. (2005). Mean curvature flow with a forcing term in Minkowski space. Calc. Var. Partial Differ. Equ. 25(2): 205--246 · Zbl 1094.53060 · doi:10.1007/s00526-005-0351-8 [2] Andrews B. (1994). Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2(2): 151--171 · Zbl 0805.35048 · doi:10.1007/BF01191340 [3] Chow B. and Gulliver R. (2001). Aleksandrov reflection and geometric evolution of hypersurfaces. Comm. Anal. Geom. 9(2): 261--280 · Zbl 1004.53048 [4] Chou K. and Wang X. (2000). A logarithmic Gauss curvature flow and the Minkowski problem. Ann. Inst. Henri Poincaré Anal Non Linéaire 17(6): 733--751 · Zbl 1071.53534 · doi:10.1016/S0294-1449(00)00053-6 [5] Gerhardt C. (1990). Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32: 299--314 · Zbl 0708.53045 [6] Hamilton R. (1982). Three manifolds with positive Ricci curvature. J. Differ. Geom. 17: 255--306 · Zbl 0504.53034 [7] Huisken G. (1984). Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1): 237--266 · Zbl 0556.53001 [8] Huisken G. (1987). The volume preserving mean curvature flow. J. Reine Angew. Math. 382: 35--48 · Zbl 0621.53007 · doi:10.1515/crll.1987.382.35 [9] Huisken, G., Polden, A.: Geometric Evolution Equations for Hypersurfaces. Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, vol. 1713, pp. 45--84. Springer, Berlin (1999) · Zbl 0942.35047 [10] Krylov, N.: Nonlinear Elliptic and Parabolic Equations of the Second Order. D. Reidel Publishing Company (1987) · Zbl 0619.35004 [11] Liu Y. and Jian H. (2007). Evolution of hypersurfaces by mean curvature minus extermal force field. Sci. China Ser A 50: 231--239 · Zbl 1120.35019 · doi:10.1007/s11425-007-2077-x [12] McCoy J. (2003). The surface area preserving mean curvature flow. Asian J. Math. 7(1): 7--30 · Zbl 1078.53067 [13] McCoy J. (2004). The mixed volume preserving mean curvature flow. Math. Z. 246(1--2): 155--166 · Zbl 1062.53057 · doi:10.1007/s00209-003-0592-1 [14] Schnürer O. and Smoczyk K. (2002). Evolution of hypersurfaces in cental force fields. J. Reine Angew Math. 550: 77--95 · Zbl 1019.53033 · doi:10.1515/crll.2002.075 [15] Tso K. (1985). Deforming a hypersurface by its Gauss--Kronecker curvature. Comm. Pure Appl. Math. 38(6): 867--882 · Zbl 0612.53005 · doi:10.1002/cpa.3160380615 [16] Urbas J.I.E. (1991). An expansion of convex hypersurfaces. J. Differ. Geom. 33(1): 91--125 · Zbl 0746.53006 [17] Zhu, X.: Lectures on mean curvature flows. AMS/IP Studies in Advanced Mathematics, vol. 32. American Mathematical Society. International Press, Providence (2002) · Zbl 1197.53087