The structure of area-minimizing double bubbles
This remarkable paper addresses the possible configurations of minimizing \((n-1)\) dimensional area \(A_n(v_1,v_2)\) that enclose two disjoint open sets of prescribed volumes \(v_1,v_2\) in \(\mathbb{R}^n\) (double bubble). The author establishes that for each \((n\geq 3)\), the functional \(A_n(v_1,v_2)\) is strictly concave on every line in \([0,\infty)^2\). This result leads to striking new information on the topological character of minimizers; an important lemma in its proof is a symmetry property, based on an idea of Brian White, that has independent interest. The reviewer calls attention to a more recent work ``Proof of the double bubble conjecture'', joint work of the author with Morgan, Ritoré and Ros, in which the present contributions are utilized in providing the first complete proof of that long standing conjecture: in \(\mathbb{R}^3\) the unique perimeter-minimizing double bubble consists of three spherical caps meeting along a common circle at \(120^\circ\) angles.
scientific article; zbMATH DE number 1228525
scientific article