Symmetric Convex Sets with Minimal Gaussian Surface Area

Let Ω ⊆ R have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = Ax be the second fundamental form of ∂Ω at x, i.e. A is the matrix of first order partial derivatives of the unit normal vector at x ∈ ∂Ω. For any x = (x1, . . . , xn+1) ∈ R, let γn(x) = (2π)−n/2e−(x 2 1+···+x 2 n+1. Let ‖A‖ be the sum of the squares of the entries of A, and let ‖A‖2→2 denote the `2 operator norm of A. It is shown that if Ω or Ω is convex, and if either ∫ ∂Ω (‖Ax‖ − 1)γn(x)dx > 0 or ∫ ∂Ω ( ‖Ax‖ − 1 + 2 sup y∈∂Ω ‖Ay‖2→2 ) γn(x)dx < 0, then ∂Ω must be a round cylinder. That is, except for the case that the average value of ‖A‖ is slightly less than 1, we resolve the convex case of a question of Barthe from 2001. The main tool is the Colding-Minicozzi theory for Gaussian minimal surfaces, which studies eigenfunctions of the Ornstein-Uhlenbeck type operator L = ∆− 〈x,∇〉+ ‖A‖ + 1 associated to the surface ∂Ω. A key new ingredient is the use of a randomly chosen degree 2 polynomial in the second variation formula for the Gaussian surface area. Our actual results are a bit more general than the above statement. Also, some of our results hold without the assumption of convexity.