By Bethuel, Huisken, Müller and Steffen

ISBN-10: 3540659773

ISBN-13: 9783540659778

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Additional resources for Calculus of Variations and Geometric Evolution Problems

Example text

Remark : Recently, in a joint work with B. Helffer, we have been able to show that the estimate in Proposition 15 is in some sense optimal. More precisely, we proved (see [BH]) that ttiere is some constant #1 > 0 such that the Morse Index of v, is less than ~lldl 2. 2. Variational methods Here we come back to the general case, without symmetries. In view of the previous analysis one might expect to find more solutions as d increases. consistent with the following. This hope is also P r o p o s i t i o n 17.

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. The computations have been carried out in detail for f = - U in [34], [35] and [4].

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Calculus of Variations and Geometric Evolution Problems by Bethuel, Huisken, Müller and Steffen


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