By Bethuel, Huisken, Müller and Steffen

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Has d i s t ( a i , a i ) _> #2, Vi#j, where #2 is some constant. Finally arguing as previously we may prove that the map q5 : E " ~ 2 , u ---* O(u) = ( a l , . . , a a ) is ~-almost continuous, form some r/--~ 0 as ~ --* 0. , bd(O)), and 9 o 7* is not contractible in 51. This completes the proof. 4. e. r l ( Z ~ ) = 0). On the other hand, by Proposition 17, 7h(E") # 0, for a = ~, + X0. Hence there is a critical value C > ~e + X0 hence a non minimizing solution to GL~. A third solution can then be constructed using a mountain-pass type argument (see [AB2]).

2 are analytically hard to control, compare the dependance of the main result in [35] on these terms. For harmonic mean curvaturc flow and flows of similar structure Andrews derives an optimal convergence result for hypersurfaces having sufficiently positive principal curvatures in relation to the ambient curvature, [~l]. In particular, he shows that such flows contract convex hypersurfaces in manifolds of positive sectional curvature to a point and gives a new argument for the classical 1/4-pinching theorem.

Llcrc dlz is the induced mcasure on the hypcrsurface and A is the Laplace-Beltrami operator with respect to the time-dependent induced metric on the hypersurface. Notice that - A f - f(lAI 2 + l~ic(~, ~)) = J f is the Jacobi operator acting on f , as is wellknown from the second variation formula for the area. P r o o f . A4'* and {y~} near F(p) in N. Arranging coordinates at a fixed point p such that g~j(p) = ~,j, (O/Ox')gjk(p) = O, ~,a(F(p)) = ~,,a, (O/Oy~')ga6(F(P)) = 0 all identities are straightforward consequences of the definitions and the Gauss-Weingarten relations.