The positive mass conjecture for the Gauss–Bonnet–Chern mass

Let $n$ and $q$ be integers with $n\geq 3$ and $1\leq q<n/2$. Let $(M^n,g)$ be asymptotically flat of order $\tau>(n-2q)/(q+1)$, with integrable nonnegative $q$-th Gauss–Bonnet curvature $L_{(q)}$. Its $q$-th Gauss–Bonnet–Chern mass is

$$\mathrm{m}_q=c_q(n)\lim_{r\to\infty}\int_{S_r}P_{(q)}^{ijkl}g_{jk,l}\nu_i\,dS_r,$$

where $c_q(n)=(n-2q)!/[2^{q-1}(n-1)!\omega_{n-1}]$. Positive mass conjecture for the Gauss–Bonnet–Chern mass. The mass $\mathrm{m}_q$ is nonnegative. Moreover, if it is zero, then $(M,g)$ is isometric to $(\mathbb{R}^n,\delta)$.

This is presented as the Gauss–Bonnet–Chern analogue of the positive mass conjecture. The supplied text does not give a resolution status for the general statement, although it discusses proofs for graphical manifolds with flat normal bundle.