具有超临界指数的$L_p$ -Minkowski问题
The $L_p$-Minkowski problem with super-critical exponents
摘要：
$L_p$ -Minkowski问题研究了在给定$p$ -面积测度下，$\mathbb{R}^{n+1}$中闭凸超曲面的存在性.它扩展了经典的闵可夫斯基问题，并包含了几个重要的几何和物理应用。解的存在性在亚临界情况下已得到$p>-n-1$，但问题仍然广泛开放的超临界情况下$p<-n-1$。在本文中，我们引入新的想法来解决所有的超临界指数的问题。我们证明中的一个关键因素是基于椭圆体拓扑空间同调计算的拓扑方法。
英文摘要：
The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains widely open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids.
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