Mean Curvature Ow of Graphs in Σ1 × Σ2

Abstract

" Let \u03a3_1 and \u03a3_2 be m and n dimensional Riemannian manifolds of constant curvature respectively. We assume that w is a unit constant m-form in \u03a3_1 with respect to which \u03a3_0 is a graph. We set v = \u2329e_1 \u2227 \u2026 \u2227 e_m, \u232a), where {e_1, \u2026, e_m} is a normal frame on \u03a3_t. Suppose that \u03a3_0 has bounded curvature. If v(x, 0) \u2265 v0 > \\frac{\\sqrt{p}}{2} for all x, then the mean curvature flow has a global solution F under some suitable conditions on the curvatrue of \u03a3_1 and \u03a3_2."