Let $ (M^4,g)$ be a Ricci-flat Riemannian manifold asymptotic to the Taub-NUT metric. That is, there exists a circle fibration $ \pi:M\backslash K \to \mathbb R^3\backslash B(0,R)$ which is the Hopf fibration if restricted on $ S^2$ . Moreover, the metric $ g$ satisfies $ $ g=h+O(r^{-\tau}) ,\quad \nabla ^{h,k} g=O(r^{-\tau-k}) $ $ for some $ \tau>0$ and any integer $ k \ge 1$ , where $ h$ is the Taub-NUT metric and $ r=\sqrt{x_1^2+x_2^2+x_3^3} \circ \pi$ .

Now we define $ D_{s}=\{x\in M \mid r(x) \le s\}$ and consider the Hirzebruch signature formula: $ $ \tau(M)= \frac{1}{12 \pi^2} \int_{D_{s}} |W_+|^2-|W_-|^2\, dV+I(s)+\eta(\partial D_s) $ $ where $ I(s)$ involves the boundary integral of the second fundamental form and $ I(s) \to 0$ if $ s \to \infty$ .

Can we prove that $ \lim_{s \to \infty}\eta(\partial D_s)=\frac{2}{3} $ ?