3 problems

Positive mass conjecture for arbitrary asymptotically hyperbolic ends. The mass functional of the end $\mathcal{E}$ is timelike future-directed or zero.

Exponential wave trace remainder conjecture. The remainder $A$ is smooth on $(0,\infty)$, and there exist $C,k>0$, depending only on $n$, $k_1$, and $k_2$, such that

Let $M_j^m$ be a sequence of asymptotically hyperbolic Riemannian manifolds with scalar curvature $R\geq -m(m-1)$ and mass tending to zero. Assume that $M_j^m$ contains no closed i…