108 problems

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 $(M^n,g)$ 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…

Finite-volume volume conjecture. The volume satisfies

Volume-entropy conjecture. If

Let $M$ be a closed $3$-manifold that admits a constant-curvature metric $h$. Here $Y(M)$ denotes the Yamabe invariant of $M$, and $E(h)$ denotes the normalized Einstein-Hilbert en…

Conjecture on negative transverse curvature. There is a Riemannian metric $g$ on $M$ such that the function $\chi_{g,\bot H}$ is everywhere negative.

Hamiltonian volume conjecture. The rescaled volume satisfies

WYD scalar-curvature conjecture. There exists $\varepsilon>0$ such that, for every $p\in I$, the scalar curvature of the WYD$(p)$ metric is a Schur-increasing function.

Commutative alpha-geometry conjecture. Suppose $n>2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then $\operatorname{Scal}_p$…

Petz's conjecture. The scalar curvature of the BKM metric is a Schur-increasing function; equivalently,

Let $D_1$ and $D_2$ be states, and write $D_1\prec D_2$ when $D_1$ is more mixed than $D_2$ in the majorization order. Let $\operatorname{Scal}(D)$ denote the scalar curvature of t…

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…

A closed symplectically aspherical manifold is a closed manifold whose symplectic form vanishes on every spherical homology class. The symplectically aspherical positive scalar cur…

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 $M$ 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 $C$-…

Boundary curvature-integral conjecture. One has

Generalized LeBrun equality. The equality above should hold for any compact complex $n$-manifold, at least when $K_M$ is big.

Akutagawa's radiality conjecture. This infimum is achieved by a function radial in the $\mathbb{R}^{m_2}$ variable.

Yamabe product conjecture. The Yamabe constant $Y(g(T))$ is minimized by $u_{n,T}$, and

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 $X$ and $X_i$ be closed oriented Riemannian manifolds, and let $W_{\alpha_i,\beta_i,\lambda_i}\colon X_i\to X$ denote Riemannian $(\alpha_i,\beta_i,\lambda_i)$-cobordisms, mean…

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…