论文标题
双曲线空间中的Michael-Simon类型不平等
Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows
论文作者
论文摘要
在本文中，我们首先建立并验证了Michael-Simon不平等的新型尖锐的双曲线版本，用于夸张空间中的平均曲率$ \ MATHBB {H}^{n+1} $，基于本地限制的逆弯曲流，由Brendle，Guan和Li引入了$ M $ H $ -Conve and $ h $ $ $ fcortial $ h $ faction in $λ^{'}（r）= \ rm {cosh} $$ r $。特别是，当$ f $恒定时，（0.1）与布伦德尔（Brendle），悬挂和王（Wang）所说的Minkowski类型不等式相吻合。此外，我们还建立并确认了$ k $ - the Mathbb {h}^{n+1} $中的$ k $ - th平均曲率的新夏普·西蒙（Mather-simon）的不平等，但通过Brendle-guan-li的流量，前提是$ m $是$ h $ -convex和$ω$是$ m $ $ m $。特别是，当$ f $恒定而$ k $奇怪时，（0.2）正是Hu，Li和Wei证明的加权Alexandrov-Fenchel不平等。
In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li, provided that $M$ is $h$-convex and $f$ is a positive smooth function, where $λ^{'}(r)=\rm{cosh}$$r$. In particular, when $f$ is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang. Further, we also establish and confirm a new sharp Michael-Simon inequality for the $k$-th mean curvatures in $\mathbb{H}^{n+1}$ by virtue of the Brendle-Guan-Li's flow, provided that $M$ is $h$-convex and $Ω$ is the domain enclosed by $M$. In particular, when $f$ is of constant and $k$ is odd, (0.2) is exactly the weighted Alexandrov-Fenchel inequalities proven by Hu, Li, and Wei.
