论文标题
在加权圆柱体中具有边界的超曲面的独特结果和外壳特性
Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders
论文作者
论文摘要
对于可能具有边界的Riemannian歧管$ M $，我们考虑了Riemannian产品$ M \ times \ Mathbb {r}^k $，具有平稳的正功能，可以加权Riemannian措施。在这项工作中，我们表征具有非空边界的抛物线超曲面，并包含在$ M \ times \ Mathbb {r}^k $的某些区域内，具有合适的权重。我们的结果包括加权圆柱体中的半空间和伯恩斯坦型定理。我们还推广到此设置一些经典特性，这些特性是根据其边界的位置将紧凑的最小超表面限制到欧几里得空间的某些区域。最后，我们显示了应用陈述的有趣情况，其中一些与平均曲率流的奇异性有关。
For a Riemannian manifold $M$, possibly with boundary, we consider the Riemannian product $M\times\mathbb{R}^k$ with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with non-empty boundary and contained within certain regions of $M\times\mathbb{R}^k$ with suitable weights. Our results include half-space and Bernstein-type theorems in weighted cylinders. We also generalize to this setting some classical properties about the confinement of a compact minimal hypersurface to certain regions of Euclidean space according to the position of its boundary. Finally, we show interesting situations where the statements are applied, some of them in relation to the singularities of the mean curvature flow.
