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Riemannian geometry a beginners guide - Morgan F.

Morgan F. Riemannian geometry a beginners guide - Jones and Bartlett, 1993. - 121 p.
ISBN 0-86720-242-4
Download (direct link): riemanniangeometry1993.djvu
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area S ^ a[<F(C)]2,
with equality for C = C0.
Almgren’s methods [Aim, esp. Section 9] using geometric measure theory show that such an optimal isoperimetric curve exists.
94 CHAPTER 10
In the plane such curves have a nice characterization. Let 'VL be a 90° rotation of 4>, so
where n is the unit normal obtained by rotating the unit tangent T 90° counterclockwise. The dual norm W* is defined by
= sup{v • w: < 1},
so |u • w| < 'L(w). The optimal isoperimetric curve is simply the
boundary of the unit 0*-ball or “Wulff shape” (see [Wu] or [Tl]). Here we sketch a short new proof, based on Schwartz symmetriz-ation, as recently used by Brothers [Br] and Gromov [BG, Section 6.6.9, p. 215] and earlier by Knothe [K]. The same result and proof hold for optimal isoperimetric hypersurfaces in all dimensions.
10.6. Theorem. Let W be a norm on R2. Among all curves enclosing the same area, the boundary of the unit 'L*-ball B (Wulff shape) minimizes fdB ^(n).
Proof sketch. Consider any planar curve enclosing a region B' of the same area as B. Let/be an area-preserving map from B' to B carrying vertical lines linearly to vertical lines. Then det Df= 1 and Df is triangular:
Since det Df = ab = 1, div/= a + b ^ 2. Hence
c
> / • n = div/> 2 area B' = 2 area B,
dB' B'
with equality if B' = B.
Remark. Careful attention to the inequalities in the proof recovers the result of J. Taylor [T2] that the Wulff shape is the unique minimizer among measurable sets [BrM].
GENERAL NORMS 95
Figure 10.3. The unit <I>-ball B. Any slice A by a plane P through the origin has a vertex ax that is not a vertex of B.
In general dimensions, optimal isoperimetric curves are not well understood. Obvious candidates are planar Wulff shapes. The following new theorem says, however, that optimal isoperimetric curves are not generally planar.
10.7. Theorem. For some convex norms in R3, an optimal isoperimetric curve is nonplanar.
Proof. Define by taking the unit <F-ball to be the centrally symmetric polyhedron B of Figure 10.3. In any plane P, which we may translate to pass through the origin, the Wulff shape S with boundary C maximizes (area S)/<F(C)2. We will show that this ratio is larger for some nonplanar curve C.
The slice A of the unit <F-ball B by the plane P must be polygonal. At least one vertex «i is not a vertex of B. The vertex a1 must lie on an edge of B, with vertices b0, hi- The Wulff shape S is the polygon formed dual to A, rotated clockwise 90°. (See Figure
96 CHAPTER 10
S
Figure 10.4. In the plane P, the Wulff crystal S is the polygon dual to A, rotated 90° clockwise.
10.4.) If Ci denotes the edge dual to ax (rotated 90°), then Ci points in the «i direction. Its distance from the origin is l/|tfi|.
Let C' be the polygon in R3 obtained from C by replacing Ci by two segments in the directions b0, hi, in the order that keeps the projection PC' of C' onto P out of the interior of C. (See Figure 10.5.) Then <F(C') = <F(C). Let S' be an area-minimizing surface bounded by C'. Then
Remark. By approximation one obtains examples that are also smooth and elliptic.
For length, optimal isoperimetric curves are circles of constant curvature. For general <h, the generalized curvature at least satisfies an inequality.
10.8. Lemma. For a C2 optimal isoperimetric curve C0, the generalized <b-curvature vector satisfies
area S' > area PS' > area S.
Consequently,
area S' area S <h(C')2 <h(C)2
(1)
GENERAL NORMS 97
Figure 10.5. Obtain C' from С by replacing C\ with segments in the directions b0, b\. Then Ф(С') = Ф(С), but area S' > area S.
Remarks. For the case where $ is length and C0 is the unit circle, (1) says |k| ^ 1. The smoothness hypothesis on C0 is unnecessary; still the conclusion implies that C0 is C1,1. If C0 bounds a unique smooth area-minimizing surface S0 with n the inward normal to Co along So, D2<&(k) actually must be a constant multiple of n. In particular, a planar optimal isoperimetric curve has constant generalized curvature:
|Z)2$(k)| = K.
Proof. Let/: [0, a] —*■ R" be a local arc length parameterization of C0. Consider compactly supported variations 8f. Then
0 ^ S(area S - a$(C)2)
> - | |S/| ds + 2a$(Co) J D2®(k) • 8fds
by 10.4. Therefore
|D2<D(k)| <-------------=
2 a<F(C0) 2 area S0
A norm Ф is called crystalline if the unit Ф-ball is polytope.
98 CHAPTER 10
Figure 10.6. Length-minimizing networks meet in threes at equal angles of 120°.
10.9. Conjecture. If is crystalline, then an optimal isoperimetric curve is a polygon.
10.10. <E>-minimizing networks. A network N is a finite collection of line segments. Given a norm <I> and a finite set of boundary points in R", we seek a ^-minimizing network connecting the points. For the case where 4> is length (the generalized “Steiner” or “Fermat” problem [Fer, 1638, p. 153], [SI, 1835], [S2, 1837], [JK, 1934]), such networks meet only in threes at equal 120° angles (or in twos at boundary points at angles of at least 120°), as shown in Figure 10.6. Soap film strips behave similarly in their quest to minimize area, as shown in Figure 10.7 (see also [CR, pp. 354-361, 392]).
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