Proof of the double bubble conjecture in \(\mathbb{R}^4\) and certain higher dimensional cases. (Q1880025)

The standard double bubble is, by now, one of the most familiar and controversal objects in the `\(n\)-dimensional soap bubble domain'. As most scientists know, the standard double bubble consists of three spherical caps meeting at 120 degree angles, and it is conjectured as the least-area hypersurface that encloses two given volumes in \(\mathbb R^n\). The case \(n=3\), solved in 2000 by another team of four mathematicians, is now a very well known result. The four authors of the present study prove the conjecture for \(n=4\), and open some important directions towards higher dimensions. The arguments are well organized and easy to follow, making this report accessible to a large audience of mathematicians, including graduate students.