Symmetric convex sets with minimal Gaussian surface area

Abstract

Let Ω ⊆ R n+1 have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = A x 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. Let A 2 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 c is convex, and if eitherthen ∂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.

Symmetry of minimizers of a Gaussian isoperimetric problem

“…We estimate the right-hand-side of (22) in the following way. We estimate the first term by Young's inequality…”

“…We mention also a somewhat similar result by Latala and Oleszkiewicz [26,Theorem 3] who proved that the symmetric strip minimizes the Gaussian perimeter weighted with the width of the set among convex and symmetric sets with volume constraint. For the standard perimeter the problem is more difficult as a simple energy comparison shows (see [22]) that when the volume is exactly one half, the two-dimensional disk and the three-dimensional ball have both smaller perimeter than the symmetric strip in dimension two and three, respectively. Similar difficulty appears also in the isoperimetric problem on sphere for symmetric sets, where it is known that the union of two spherical caps does not always have the smallest surface area (see [4]).…”

“…Similar difficulty appears also in the isoperimetric problem on sphere for symmetric sets, where it is known that the union of two spherical caps does not always have the smallest surface area (see [4]). However, it might still be the case that the solution of the problem is a cylinder B k r × R n−k , or its complement, for some k depending on the volume (see [22,Conjecture 1.3]). Here B k r denotes the k-dimensional ball with radius r. At least the results by Heilman [21,22] and La Manna [25] seem to indicate this.…”

“…However, it might still be the case that the solution of the problem is a cylinder B k r × R n−k , or its complement, for some k depending on the volume (see [22,Conjecture 1.3]). Here B k r denotes the k-dimensional ball with radius r. At least the results by Heilman [21,22] and La Manna [25] seem to indicate this.…”

Convexity of $λ$-hypersurfaces

“…In the curve case, Guang [Gua18] proved that any smooth embedded 1-dimensional λ-hypersurface (or λ-curve) is either a straight line or a circle if λ ≥ 0, which generalized Abresch and Langer's result. For the higher dimensional case, Heilman [Hei17] proved that convex n-dimensional λhypersurfaces are generalized cylinders. However, when λ < 0, Chang [Cha17] showed that for certain λ < 0, there are some closed embedded mean convex λ-curves other than circles.…”