论文标题
双曲线空间中局部约束的曲率流和几何不平等现象
Locally constrained curvature flows and geometric inequalities in hyperbolic space
论文作者
论文摘要
在本文中，我们首先研究了双曲线空间中高曲面的局部约束曲率流，这是由Brendle，Guan和Li [7]引入的。该流量保留了$ m $ the QuermassIntegral，并减少$（M+1）$ TH QuermassIntegral，因此流量的收敛性产生了敏锐的Alexandrov-Fenchel型不平等现象，在多余的空间中。 [7]中已经研究了一些特殊情况。在本文的第一部分中，我们表明沿流动曲面的H-串联度被保留，然后将H-Convex Hypersurfaces的流量平滑收敛。 We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $ 0 \ leq m \ leq k \ leq n $）。特别是，我们对2015年GE，Wang和Wu提出的猜想给出了肯定的答案。 在本文的第二部分中，我们使用移动的主曲线引入了新的局部约束曲率流。在H-凸度的背景下，这是自然的。 We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue布伦德尔（Brendle），洪（Hung）和王（Wang）[8]不平等。
In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.
