Download e-book for kindle: Existence and regularity of minimal surfaces on Riemannian by Jon T. Pitts

Geometry Topology

By Jon T. Pitts

Mathematical No/ex, 27

Originally released in 1981.

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This need not always be so, as illustrated by the following example, due to Almgren [AF2, p. 15-8]. Here M is diffeomorphic to s2 metrized as a "three-legged starfish" (figure 5). critical path M • ~ ~ , but is One is illustrated, and we have graphed (length vs. t) below the manifold. All curves ~(t) 21 ----- ------ .......... ....... 0 1 c I Figure 5. c) Critical path ¢ on the three-legged starfish are simple and closed. have two components. The critical curve shaped like a figure eight point of intersection.

22] f: S - Y, and v ::::_ 1. v relative to S if and only if is of class p E S, there exist a neighborhood and a map Suppose g : U - Y of class v such that We may also define the differential of f U of p in Rn f I (Sn U) = g I (Sn U). relative to S at p, Df(p) = Dg(p) I Tan(S,p). (4) Submanifolds of Rn. ============ == = (Cf. 19]) Let m,v 2::. 1. , 50 1 m-l w 0 cp R +) Wn M and W n image (cp). In case xR M is a submanifold with boundary, then the boundary of M, oM, is the set of all points for some map M ~ 0M ~ p E M such that as described above.

Here we discuss the ideas behind one such property of varifolds: Let almost minimizing. M be a compact manifold and let open subset of in M• A varifold U if, roughly speaking, arbitrarily closely in V U be an is almost minimizing V may be approximated U by integral cycles which are themselves almost locally area minimizing in U . 1. 1(6) we have already seen a simple example of a critical surface which is an almost minimizin varifold. In the notation of that example, we define a sequence of integral cycles: T.

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