Student Analysis Seminar

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Bernstein’s problem on minimal surfaces

Speaker: Andrew Lee, NYU Courant

Location: Warren Weaver Hall 512

Date: Monday, October 5, 2026, 12:30 p.m.

Synopsis:

Minimal surfaces—local minimizers of area—are amongst the most classical objects of mathematics: 1-dimensional minimal surfaces are geodesics, and 2-dimensional minimal surfaces model the behavior of soap bubbles. A very natural problem is to classify minimal surfaces in the codimension 1 Euclidean case (\(n\)-dimensional minimal surfaces in \(\mathbb{R}^{n+1}\)). Geometric intuition tells us that hyperplanes are minimal surfaces, but in 1917 S. Bernstein posed the converse question: are \(n\)-dimensional hyperplanes the only minimal surfaces in \(\mathbb{R}^{n+1}\)? I will explain the answer to this question, which for deep geometric and analytic reasons (and the quadratic formula) turns out to be yes for \(n \leq 7\) and no for \(n \geq 8\).