Mathematician unleashes 'a wave of new results' in geometric analysis: "If you drive into a parking garage and go up a level, that spiral ramp is part of a helicoid," he says. "And one of the things we were able to show was that every embedded minimal surface could be built out of these things. So the minimal surface either looks like a nice flat thing where the area is bounded, or it looks exactly like one of these double spiral staircases..."
They have been able to prove, he says, that of the infinite number of singularities that could possibly affect a surface through this curvature flow, only two types are stable enough to survive in reality. "If you were to wiggle your surface ever so slightly, in fact only two are stable of the infinitely many that are possible," Minicozzi says. "So if you are trying to understand [mean curvature flow] and you have to deal with all of these cases, it's much better to have to deal with two cases than an infinite number."
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