111 problems

Let $n\ge 3$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold with positive scalar curvature. If, for some (equivalently any) $x_0\in M$,…

For every integer $n\ge 3$, there exists a constant $\varepsilon_n\in(0,1)$ such that, for every connected, closed, smooth Riemannian manifold $(M^n,g)$, if…

Let $(M,g)$ be a closed Riemannian $3$-manifold, let $R_g$ denote its scalar curvature, and let $h(g)$ denote its volume entropy. Agol–Storm–Thurston conjecture. If … then … This c…

For every complete, connected, noncompact Riemannian $3$-manifold $(M,g)$ without boundary satisfying $\mathrm{Scal}_g\ge 6$, the conjugate radius satisfies…

The conjecture asserts a quantitative width bound for foliated positive-scalar-curvature geometry: under the appropriate completeness, foliation, boundary, index, rank, and leafwis…

Let $g$ be a smooth complete metric on $\mathbb R^n$, where $n\geq 3$, and let $g_{\mathrm{euc}}$ denote the Euclidean metric. Assume that … and … as $x\to\infty$. Gromov's Euclide…

Let $M$ be a closed smooth manifold, and let $g$ be an $L^\infty$-metric on $M$, meaning that for some smooth reference Riemannian metric $g_0$ and constant $\Lambda>0$,…

Let $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced…

Gromov's upper bound conjecture. There exists a constant $C(M,\sigma)$ such that for every such $X$,

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindent\text{in}cmathbb{R}$, let … be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic…

Let $M$ be an orientable $n$-manifold equipped with a complete Riemannian metric of positive scalar curvature. For a complete Riemannian metric on $M$, say that it has at most $C$-…

Let $M$ be a manifold of dimension $m$, and let $T^m$ denote the $m$-dimensional torus. Generalized Geroch conjecture. There is no complete metric of positive scalar curvature on ……

Hamilton's conjecture. The metric $g(t)$ converges to a metric of constant scalar curvature as $t\to\infty$.

Yamabe's conjecture. There exists a metric $g$ on $X$ conformally equivalent to $g_0$ and having constant scalar curvature.

Boundary curvature-integral conjecture. One has

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 $M$ be a manifold diffeomorphic to $S^3$, equipped with a Riemannian metric. An embedded minimal two-sphere in $M$ is an embedded minimal submanifold diffeomorphic to $S^2$. S.…

Let $C_s,v_0>0$ and $n\geq 3$. Let $(M^n,g_0)$ be a complete non-compact Riemannian manifold. Assume that … … … and $\mathcal{R}(g_0)\geq 0$. Positive scalar-curvature rigidity con…

Let $(M^3,g)$ be a three-dimensional complete manifold with Ricci curvature $\operatorname{Ric}\geq 0$, and let $B_p(R)$ denote the geodesic ball of radius $R$ centered at $p\in M$…

Marques–Neves' width stability conjecture. Then $(\mathbb S^3,g_k)$ converges in the Sormani–Wenger intrinsic flat sense to the round unit sphere $(\mathbb S^3,g_{rd})$. The conjec…

Gromov's conjecture. If no bubbling occurs along the sequence, or if the bubbles forming along the sequence can be cut out, then a subsequence of $M_j$ converges in some sense to t…

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 $(M^n,g)$ be a complete noncompact Riemannian manifold with $n\ge5$. Write $R_g$ for its scalar curvature, and let $\mathcal S^2$ denote the $L^2$-scalar curvature functional.…

Let $(M,h_0)$ be a closed hyperbolic manifold of dimension $n\geq 3$, and let $N$ be a closed $n$-manifold with Riemannian metric $g$ whose scalar curvature satisfies…

Let $M$ be a compact, locally irreducible Riemannian manifold that is rigidly scalar-flat, meaning that it admits a scalar-flat metric but no metric of positive scalar curvature. I…