Given a contact 3-manifold, we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

中文翻译：

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Ricci曲率，Reeb流动和接触3歧管

给定一个接触3流形，我们考虑何时可以将给定函数实现为接触结构的Reeb矢量场的Ricci曲率的问题。我们将使用拓扑工具来表明，每个允许的函数都可以实现为奇异度量的Ricci曲率，这是远离度量零集的诚实兼容度量。但是，我们将看到解决这种奇异性取决于接触拓扑数据，并且尚未完全理解。