17 problems

Let $M_j$ be asymptotically flat $n$-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either no boundary or boundary…

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 $\Omega_j$ 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…

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 $M_i$ 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 $M_j$ be a sequence of Riemannian manifolds diffeomorphic to a $3$-torus such that … where … If the scalar curvature $R(M_j)$ satisfies $R(M_j)\geq -\frac{1}{j}$, then Sormani'…

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…

Let $M_j^3$ be a sequence of three-dimensional Riemannian manifolds. Write ${\rm Scal}_j$ for the scalar curvature, $\operatorname{Vol}(M_j)$ for their volume, and…

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

Let $\mathcal{M}$ be a subclass of asymptotically flat three-dimensional Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces, with either…

Intrinsic-flat noncancellation conjecture. A converging sequence of three-dimensional Riemannian manifolds with positive scalar curvature, a uniform lower bound on volume, and no i…