abstract:

Let $\Omega\subset\Bbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_x$ be the second fundamental form of $\partial\Omega$ at $x$, i.e., $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in\partial\Omega$. For any $x=(x_1,\ldots,x_{n+1})\in\Bbb{R}^{n+1}$, let $\gamma_n(x)=(2\pi)^{-n/2}e^{-(x_1^2+\cdots+x_{n+1}^2)/2}$. Let $\|A\|^{2}$ be the sum of the squares of the entries of $A$, and let $\|A\|_{2\to 2}$ denote the $\ell_{2}$ operator norm of $A$.

It is shown that if $\Omega$ or $\Omega^c$ is convex, and if either $$ \int_{\partial\Omega}\Big(\big\|A_x\big\|^2-1\Big)\gamma_n(x)\,dx>0\quad\textrm{or}\quad\int_{\partial\Omega} \Big(\big\|A_x\big\|^2-1+2\sup_{y\in\partial\Omega}\big\|A_y\big\|_{2\to 2}^2\Big)\gamma_{n}(x)\,dx<0, $$ then $\partial\Omega$ 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 $L=\Delta-\langle x,\nabla \rangle+\|A\|^{2}+1$ associated to the surface $\partial\Omega$. 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.

摘要：

令$ \ Omega \ subset \ Bbb {R} ^ {n + 1} $在具有固定高斯体积的所有满足$ \ Omega =-\ Omega $的集合中，具有最小的高斯表面积。假设$ A = A_x $是$ x $处$ \ partial \ Omega $的第二基本形式，即$ A $是单位法向矢量在$ x \ in \ partial \ Omega中的一阶偏导数的矩阵$。对于任何$ x = {x_1，\ ldots，x_ {n + 1}）\ in \ Bbb {R} ^ {n + 1} $，令$ \ gamma_n（x）= {2 \ pi）^ {-n / 2} e ^ {-（x_1 ^ 2 + \ cdots + x_ {n + 1} ^ 2）/ 2} $。设$ \ | A \ | ^ {2} $为$ A $项的平方之和，并使$ \ | A \ | _ {2 \ to 2} $表示$ \ ell_ {2} $ A $的$运算符范数。

结果表明，如果$ \ Omega $或$ \ Omega ^ c $是凸的，并且$$ \ int _ {\ partial \ Omega} \ Big（\ big \ | A_x \ big \ | ^ 2-1 \ Big ）\ gamma_n（x）\，dx> 0 \ quad \ textrm {or} \ quad \ int _ {\ partial \ Omega} \ Big（\ big \ | A_x \ big \ | ^ 2-1 + 2 \ sup_ {y \ in \ partial \ Omega} \ big \ | A_y \ big \ | __ {2 \ to 2} ^ 2 \ Big）\ gamma_ {n}（x）\，dx <0，$$然后$ \ partial \ Omega $必须是圆柱体。也就是说，除了$ \ | A \ | ^ {2} $的平均值略小于$ 1 $的情况外，我们解决了2001年以来Barthe问题的凸情况。

主要工具是高斯极小曲面的Colding-Minicozzi理论，该理论研究Ornstein-Uhlenbeck类型算子$ L = \ Delta- \ langle x，\ nabla \ rangle + \ | A \ | ^ {2} + 1 $与表面$ \ partial \ Omega $相关联。关键的新成分是在第二个变化公式中针对高斯表面积使用随机选择的2阶多项式。我们的实际结果比上述陈述更为笼统。同样，我们的某些结果在没有凸度假设的情况下保持不变。