2017/05/18 by Heilman, Steven
#Computational Complexity (cs.CC) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1705.06643
Let Ω⊂ℝn+1 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)∈ℝn+1, let γn(x)=(2π)-n/2e^-(x12+⋯+xn+12)/2. Let ‖A‖2 be the sum of the squares of the entries of A, and let ‖A‖2→ 2 denote the ℓ2 operator norm of A. It is shown that if Ω or Ωc is convex, and if either ∫∂Ω(‖Ax‖2-1)γn(x)dxgt;0\qquador ∫∂Ω(‖Ax‖2-1+2supy∈∂Ω‖Ay‖2→ 22)γn(x)dxlt;0, then ∂Ω must be a round cylinder. That is, except for the case that the average value of ‖A‖2 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= Δ-⟨ x,∇ ⟩+‖A‖2+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.