The asserted double bubble theorem for S n follows. Here, one checks the Hutchings hypothesis 19.2( 2) in G 2, whence it follows for S n for large n and thence for all n. In Theorem 18.2, this relationship was used to solve the single bubble problem in G 2, to prove that lines bound regions (half-planes) of prescribed area and minimum perimeter in G 2.
How did they verify 19.2( 2) for all n in one fell swoop? They considered the Gauss plane G 2, the limit of orthogonal projections of S n, suitably normalized, as n approaches infinity (see Chapter 18). (2007) proved the Double Bubble Conjecture in S n when each region and the complement has volume fraction within. The authors prove analogous results in hyperbolic space H 3.įinally, Corneli, Corwin, et al. These results cover most of the territory where the hypothesis of 19.2 with k = 2 holds, and similar computer analysis in S 4 seems impractical. (2007) used extensive and clever computer analysis to prove the Double Bubble Conjecture in S 3 when each enclosed volume and the complement occupy at least 10% of the volume of S 3. For S 3, Cotton and Freeman (2002) used computer analysis to verify 19.2( 2) for all three regions and prove the Double Bubble Conjecture in S 3 when the prescribed volumes are equal and the exterior occupies at least 10% of S 3. It is not known whether Reichard's component-bound-free methods for R n (Section 14.0) can be generalized to S n.įor the exceptional case of S 2, Masters (1996) computer checked a variant of 19.2( 2 ) just for the largest region and then proved the Double Bubble Conjecture in S 2 for all areas. Unfortunately, hypothesis 19.2( 2), although sufficient, is not necessary, and it fails in many cases. Frank Morgan, in Geometric Measure Theory (Fifth Edition), 2016 19.4 Double Bubble Theorems in S nĭouble bubble theorems in S n to date have verified 19.2( 2) in certain cases and then used 19.3 to conclude that a perimeter-minimizing double bubble must be standard.
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