关于规定加权标量曲率和加权Yamabe流的问题

Pub Date : 2023-01-01 DOI:10.1515/agms-2022-0152
P. Ho, Jin‐Hyuk Shin
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引用次数: 1

摘要

Case引入的加权Yamabe问题是Gagliardo-Nirenberg不等式对光滑度量测度空间的推广。更准确地说,给定光滑度量测度空间(M,g,e−ξd V g,M)\left(M,g,{e}^{-\phi}{\rm{d}}){V}_{g} ,m),加权Yamabe问题在于找到与(m,g,e−ξd V g,m)\left(m,g,{e}^{-\phi}{\rm{d}}共形的另一个光滑度量测度空间{V}_{g} ,m),使得其加权标量曲率等于λ+μe−ξ/m\lambda+\mu{e}^{-\phi/m},对于一些常数μ\mu和λ\lambda,满足一定条件。在本文中,我们考虑了指定加权标量曲率的问题。我们首先证明了关于加权标量曲率的一些唯一性和非唯一性结果,然后证明了一些存在性结果。我们还估计了加权拉普拉斯算子的第一个非零特征值,即(M,g,e−ξd V g,M)\left(M,g,{e}^{-\phi}{\rm{d}}){V}_{g} ,m)。另一方面,我们证明了(M,g,e−ξd V g,M)\left(M,g,{e}^{-\phi}{\rm{d}})上共形Schwarz引理的一个版本{V}_{g} ,m)。所有这些结果都是通过使用与加权Yamabe流相关的几何流来实现的。我们还证明了加权Yamabe流的后向唯一性。最后,我们考虑加权Yamabe孤子,这是加权Yamabe流的自相似解。
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On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow
Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.
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