Symmetric convex sets with minimal Gaussian surface area
Symmetric convex sets with minimal Gaussian surface area
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DOI：
10.1353/ajm.2021.0000
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发表时间：
2017-05
影响因子：
1.7
通讯作者：
Steven M. Heilman
中科院分区：
文献类型：
--
作者：
Steven M. Heilman
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.
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.