The Mean Curvature Flow Of Convex Hypersurface In Euclidean Space

In this paper, we study the mean curvature flows of convex hypersurface in Euclidean spaces in two situations: n = 1 and n≥2. we use evolution equations of their first and second fundamental forms and the maximum principle to get somestructual results on the shape of M_ω.Specifically, we prove the following theorems.Theoremâ… Suppose 7o is a closed convex curve.Then the solutionγ(,t) ofthe evolution equation with 7o as intial curve exists only at a finite timeinterval [0,ω) .Furthermore, as t→ω, the solutionγ(,t) converges to a piont.Theoremâ…¡Let n≥2 and M is a compact without boundary n -dimensional manifold, F₀ is a smooth hypersurface immersion of Mⁿ into Rⁿ⁺¹ , the intialsurface M₀ is smooth,compact without boundary and with positive mean curvature, Suppose that F_t : Mⁿ→Rⁿ⁺¹ be a one-parameter family of smooth hypersurface immersions in Euclidean space,Then the solution F(,t) of evolution equation(where H(p,t) and v(p,t) are the mean curvature and unitnormal at ( p,t ) on M_t respectively ) exists a maximal time interval 0≤t <ω<∞.Moreover as t→ω,F(,t) converges to a piont. Furthermore, ast→ω,max_Mt |A|² becomes unbounded.Theoremâ…¢The third fundamental form satisfies the evolution equation (?)/(?)t b_ij=â–³b_ij+2|A|²b_ij - 2Hb_ijH_jpg^lp-2g_ij|â–½H|²...