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

By Bethuel, Huisken, Müller and Steffen

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One can then prove that the map 9 : E a ~ W where W is the configuration space of charged 35 vortices defined in [AB2]) is 7/-continuous for some r/ = e 7, where 0 < 7 < 1. It turns out that this p r o p e r t y is sufficient to make use of the tools of algebraic topology we have in mind. V I . 2 . 2 . S k e t c h o f t h e p r o o f o f P r o p o s i t i o n 18 First wc construct the loop 7, : S1 --~ Ea (with here a = x, + X0). The idea is of course to consider a loop 7 in E which is not contractible, and then to construct a loop in E a which has precisely the vorticcs given by 7.

For simplicity we restrict attention to closed surfaces, ie compact without boundary. hi". ukl} with latin indices i , j , k, l ranging from I to n describe the intrinsic geometry of the induced metric g on the hypersurface. If u is a local choice of unit normal for F ( . M " ) , we often work in an adapted othonormal frame u, e l , . . A4'~, 1 < i , j < n. hd '~ ~ IR and the Weingarten map W = {h}} = {9a'hki} as an operator w : T . M " --, T , 3 4 " are then given by hij = ,Q ~ e t l / , e j > -~ _ < tJ, V e l e j > .

In the one--dimensional case Grayson proved that any embedded closed curve on a 2surface of bounded geometry will either smoothly contract to a point in finite time or converge to a geodesic in infinite time, compare [25], [26] and earlier work of Gage and IIamilton in [22]. In higher dimensions it is well known that singularities will in usually occur before the area of the evolving surface tends to zero. If T < (x), as is always the case in Euclidean space, the curvature of the surfaces becomes unbounded for t --* T.

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