We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincare-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture.

中文翻译：

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通向3流形II上的矢量场的公共零的存在。解决全球难题

我们对以下关于维数为3的向量场的公共零的存在的猜想：如果$ X，Y $是在$ 3 $流形$ M $上的两个$ C ^ 1 $交换向量场，而$ U $为一个相对紧凑的开盘，使得$ X $在$ U $的边界上不消失，并且在$ U $中的Poincare-Hopf指数不消失，那么$ X $和$ Y $在$ U $内有一个共同的零。当$ X $和$ Y $属于类$ C ^ 3 $且$ X $和$ Y $沿其共线的$ Y $的每个周期性轨道部分为双曲型时，我们证明了这一猜想。我们还证明了这个猜想，仍然在$ C ^ 3 $的情况下，假设流量$ Y $留下不变的横向平面场。这些结果为猜想的$ C ^ 3 $案例提供了新的启示。