2 problems

Volume-max conjecture. For a prime $6$-net $\sigma$, its ideal neoplatonic realization maximizes volume among all ideal geodesic $2$-cycles with combinatorics $\sigma$.

Let $(M,g)$ be a globally hyperbolic spacetime satisfying the strong energy condition, and let $S\subset M$ be a future trapped surface. Volume version of Penrose's singularity the…