Nondegeneracy of positive solutions to a Kirchhoff problem with critical Sobolev growth

In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u=u^{5}, & u>0 & \text{in }\mathbb{R}^{3},\end{eqnarray*} where $a,b>0$ are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equaitons.