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By Michael Spivak

Booklet by way of Michael Spivak, Spivak, Michael

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Extra resources for A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition

Example text

In the flat case we assume furthermore that the area of (M,g ) is independent of n. Then there exist smooth diffeomor- phisms which are orientation preserving if f of M is oriented, such that a subsequence of in S C°° towards a smooth metric. If which is an isometry for all g M f *g M converges admits a symmetry then the maps f can also be chosen to be S-symmetric and to map each half of M to itself. TROMBA Before proceeding with the proof we should note that this is a compactness theorem for Riemannfs moduli space, cf.

Un (C" n,v ) v ^v~ 0 and hence always -* u (C1 ) n n,v is shor- length u (C )^a(6(p )). ^ n n, v "n,v In view of (2) this shows the equicontinuity of in the point (u |3M) p n € M. The theorem is proved. 1 and N . We consider the be some compact subset of lET variational problem (u,g) where E(u,g) -* min in the set of all pairs g is a smooth metric on M and u a map 1 N of Sobolev class H2(M,]R ) with continuous boundary values mapping C. monotonically onto u(p) €K for almost all minimizing sequence T.

R|ur|2dr 1 The assertion follows then immediately from (1). TROMBA y flat case we can shift the geodesic until we come back to y again. Therefore tric to a planar rectangle (x,0) and (x,l) parallel to itself [0,b] * [0fl] M is isome- where all points are equal and the vertical sides are iden- tified in a possibly more complicated manner. By a reasoning completely analoguous to the above one we then obtain the estimate for 1. In the next lemma we show how the condition of cohesion extends from a surface with boundary to the corresponding closed symmetric surface.

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