28 problems

Sormani's conjecture. If $g$ satisfies the min-A condition, then $(T^n,g)$ should be close to a flat torus in the intrinsic flat topology.

Intrinsic flat convergence conjecture. Under these assumptions,

Gromov–Sormani MinA scalar compactness conjecture. A subsequence converges in the volume-preserving intrinsic flat sense to a three-dimensional rectifiable limit space $M_\infty$.…

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Let $M_{i,\mathrm{ext}}$ denote the exterior region, le…

Let $(M_j,g_j)$ be future Cauchy developments with time functions $\tau_j$, and let $(M_{\infty},g_{\infty})$ be a spacetime with time function $\tau_{\infty}$. For $T$ in the rele…

Let $(M_j,g_j)$ be a sequence of complete asymptotically flat $n$-dimensional manifolds with nonnegative scalar curvature and ADM masses converging to zero. Convergence can be cons…

Let $\Omega_j$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be the…

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be t…

Let $\Omega_j(R)$ be the isoperimetric regions associated with a sequence satisfying the hypotheses of the zero mass stability conjecture, and let $B_0(R)\subset{\mathbb E}^3$ be t…

Consider $3$-manifolds with positive scalar curvature and uniformly positive $\operatorname{minA}$ invariant, where $\operatorname{minA}$ denotes the infimum of the areas of closed…

Let $\{(M_i,g_i)\}_{i=1}^\infty$ be a sequence of Riemannian $3$-tori. Assume that $\operatorname{diam}(M_i,g_i)$ is uniformly bounded above, while $\operatorname{vol}(M_i,g_i)$ an…

Let the sequence of warped product spacetimes in Example … with the taxi space portion removed. The conjecture concerns the discrepancy between Gromov–Hausdorff and Sormani-Wenger…

Let $M^3_j=(\mathbb S^3,g_j)$ be homeomorphic spheres satisfying … where $R^j$ is the scalar curvature of $M^3_j$. Here $\mathrm{width}$ is the min-max quantity defined using sweep…

Let a sequence of Riemannian manifolds be homeomorphic to a torus and have nonnegative scalar curvature, and suppose it converges in the intrinsic flat sense to a Riemannian manifo…

In the setting of spacelike slices, Sormani–Wenger intrinsic flat convergence (SWIF convergence) is the convergence of metric spaces obtained from spacelike slices of spacetimes. L…

Consider the ADM mass on smooth, rotationally symmetric $n$-manifolds, with convergence given by the pointed intrinsic flat topology. Lower-semicontinuity conjecture for ADM mass.…

Let $M^3$ be a three-dimensional Riemannian manifold, and define … Consider a sequence of such manifolds with nonnegative scalar curvature and a uniform positive lower bound for…

Let $(M'_k,g_k,x_k)$ be a sequence of pointed asymptotically flat three-manifolds with nonnegative scalar curvature and outermost minimal-surface boundaries. Suppose that … The man…

Generalized scalar-curvature conjecture. Then: (a) $M_\infty$ is a nonzero integral current space, with no cancellation without collapse; (b) $M_\infty$ is geodesic; (c) angles bet…

Almost-rigidity conjecture. A subsequence should converge intrinsically flatly to $M_0$, whose cover $\widetilde M_0$ is isometric to $\mathbb S^2\times\mathbb R$:

Almost-rigidity conjecture. A subsequence should converge intrinsically flatly to a flat torus:

Let $\Omega_0$ be a region and let $\mathcal{PM}(\Omega_0)$ denote the class of asymptotically flat extensions containing an isometric image of $\Omega_0$. Bartnik extension compac…

Fix $D>0$ and $r_0>0$. Let $\mathcal M$ be the class of asymptotically flat three-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal…

Let $F_j:M_j\to Y$ be $1$-Lipschitz, where $Y$ is compact, and suppose the $M_j$ satisfy Wenger's compactness hypotheses. After passing to a subsequence, assume … and that the push…

Let $M_j^3$ be a sequence of three-dimensional oriented manifolds satisfying … Assume also … and either that there are no closed interior minimal surfaces or that…