论文标题
在某些具有边界的加权歧管中稳定且等等的区域
Stable and isoperimetric regions in some weighted manifolds with boundary
论文作者
论文摘要
在带有光滑正函数的Riemannian歧管中，加权相关的Hausdorff测量，我们研究稳定的集合，即加权周长的二阶最小值在变化下保存加权体积。通过假设边界的局部凸度和Bakry-émery-Ricci张量的某些行为，我们通过使用从平行矢量字段构建的变形来推导稳定集的刚度属性，从而推断出稳定的集合。结果，我们将稳定的集合在某些Riemannian圆柱体中分类为$ω\ times \ times \ mathbb {r} $，并用产品权重。最后，我们还建立了唯一性结果，表明固定加权体积的加权周边的任何最小化都受水平切片$ω\ times \ {t \} $的限制。
In a Riemannian manifold with a smooth positive function that weights the associated Hausdorff measures we study stable sets, i.e., second order minima of the weighted perimeter under variations preserving the weighted volume. By assuming local convexity of the boundary and certain behaviour of the Bakry-Émery-Ricci tensor we deduce rigidity properties for stable sets by using deformations constructed from parallel vector fields tangent to the boundary. As a consequence, we completely classify the stable sets in some Riemannian cylinders $Ω\times\mathbb{R}$ with product weights. Finally, we also establish uniqueness results showing that any minimizer of the weighted perimeter for fixed weighted volume is bounded by a horizontal slice $Ω\times\{t\}$.
