具有对数凸测度的Codazzi张量$ L_p $ Aleksandrov问题的先验界和光滑解的存在性

IF 1 4区 数学 Q1 MATHEMATICS
Zhengmao Chen
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引用次数: 0

摘要

在适当的条件下,证明了具有正常截面曲率的紧致黎曼流形$ (M, g) $上具有对数凸测度的Codazzi张量$ L_p $ Aleksandrov问题光滑解的存在性。我们的证明是基于$ (M, g) $上的monge - ampontre方程的可解性,其关键因素是上述monge - ampontre方程光滑解的先验界。值得一提的是,我们的结果可以看作是将经典的欧几里得空间中的$ L_p $ Aleksandrov问题推广到具有加权测度的黎曼流形框架,我们的结果也可以看作是对Codazzi张量几何分析的一些新结果的一些尝试。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A priori bounds and existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with log-convex measure
In the present paper, we prove the existence of smooth solutions to a $ L_p $ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $ (M, g) $ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $ (M, g) $ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $ L_p $ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.
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来源期刊
CiteScore
1.30
自引率
12.50%
发文量
170
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