Local minimality of the ball for the Gaussian perimeter

Manna, Domenico Angelo La

doi:10.1515/acv-2017-0007

Cited by 8 publications

(7 citation statements)

References 19 publications

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“…On the sphere, the mean zero symmetric eigenfunctions of L which maximize the second variation of Gaussian surface area are degree two homogeneous spherical harmonics. This was observed in [Man17]. A similar observation was made in the context of noise stability in [Hei15]…”

Section: Introductionsupporting

confidence: 81%

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Symmetric convex sets with minimal Gaussian surface area

Heilman¹

2021

American Journal of Mathematics

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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.

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Section: Introductionsupporting

confidence: 81%

“…Thanks to Vesa Julin for helpful discussions, especially concerning volume preserving extensions of a function. Thanks also to Domenico La Manna for sharing his preprint [Man17].…”

Section: Discussionmentioning

confidence: 99%

Symmetric convex sets with minimal Gaussian surface area

Heilman¹

2021

American Journal of Mathematics

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“…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.…”

Section: Introductionmentioning

confidence: 94%

Symmetry of minimizers of a Gaussian isoperimetric problem

Barchiesi

Julin

2019

Probab. Theory Relat. Fields

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We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one.2010 Mathematics Subject Class. 49Q20, 60E15.

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“…In Section 2 we prove the Γ−convergence of the functional J γ s to P γ . In Section 3 we compute the first and second variation of J γ s (for the local framework see [3] or [14]). In Section 4 we prove that halfspaces are volume constrained stationary points for J γ s if and only if their Gaussian volume is 1 2 .…”

Section: Introductionmentioning

confidence: 99%

A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Rosa¹,

Manna²

2020

Preprint

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We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.

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Local minimality of the ball for the Gaussian perimeter

Cited by 8 publications

References 19 publications

Symmetric convex sets with minimal Gaussian surface area

Symmetric convex sets with minimal Gaussian surface area

Symmetry of minimizers of a Gaussian isoperimetric problem

A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

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