711 problems

Let $M$ be a smooth embedded torus in Euclidean three-space, and let its bending energy be … where $H$ is the mean curvature and $k_c$ is the bending rigidity. The Clifford torus i…

Let $\Omega$ be a compact Riemannian $3$-manifold with boundary, and consider asymptotically flat extensions satisfying the natural Bartnik boundary constraints. The infimum of the…

Penrose conjecture. If $\Sigma$ is a past apparent horizon, then

Let $\mathbb{B}^3$ be the unit ball in $\mathbb{R}^3$. A free boundary minimal annulus is a genus-zero free boundary minimal surface with two boundary components. Two such surfaces…

Let $M$ be a compact transverse Fano Sasakian manifold of dimension five, and let its Sasaki–Ricci flow be a solution of the flow equation referred to in the source. A shrinking Sa…

Let $(M^{n},g_{0})$ be a compact Riemannian manifold of dimension $n\ge 3$. Consider the Yamabe equation … where $R_{g_0}$ is the scalar curvature. Yamabe compactness conjecture. T…

Let $n\geq 2$, and let $\Sigma^{n}\subset \mathbb{R}^{n+1}$ be a complete minimal hypersurface. Calabi's conjecture. The hypersurface $\Sigma^{n}$ must be unbounded. This is the un…

Let $S_t\subset\mathbb{R}^3$ be a family of compact surfaces evolving by mean curvature flow, with generic initial data $S_0$. A singularity is spherical or cylindrical when its ta…

Let $(M^n,g,f)$, with $n\geq 4$, be a complete $n$-dimensional gradient shrinking Ricci soliton. A gradient shrinking Ricci soliton is rigid if it is isometric to a quotient of an…

For a hypersurface $\Sigma$ in Euclidean space $\mathbb{R}^{n+1}$, define its entropy by … Here the supremum is over all $t_0>0$ and $x_0\in\mathbb{R}^{n+1}$. Entropy-minimizing sp…

Let $\Sigma$ be an asymptotically cylindrical shrinker in $\mathbb{R}^{n+1}$. Ilmanen's conjecture. The shrinker $\Sigma$ must coincide with the standard cylinder. The source prese…

Let $(M,g(t))_{t\in[0,T)}$ be a solution to Ricci flow on a closed smooth $n$-dimensional Riemannian manifold, with $T$ its maximal time. Hamilton's scalar-curvature blow-up conjec…

Connectedness conjecture. The space of Anosov metrics on $\Sigma$ is connected.

The paper studies stretch maps between hyperbolic surfaces and the associated canonical geodesic laminations. Higher-dimensional extension conjecture. There should be an extension…

The Ricci flow with surgery on a 3-dimensional manifold depends on a surgery scale $\delta>0$, which can be chosen arbitrarily small. Perelman's conjecture. As $\delta$ tends to ze…

Type I scalar-curvature conjecture. Finite-time singularities of Ricci flows on Kähler manifolds should be Type I at least in the sense of scalar curvature.

Perelman's ancient oval conjecture. Every compact ancient $\kappa$-solution must either be a quotient of a family of shrinking round spheres or be isometric to a so-called ancient…

Let $M$ be a compact $3$-manifold admitting a metric with negative sectional curvature, and start the normalized positive cross curvature flow at such a metric. Chow–Hamilton conje…

Ecker's monotonicity conjecture. In the case of mean curvature flow for compact embedded hypersurfaces $M_t$ in $\mathbb{R}^{n+1}$, the entropy satisfies

Let $A$ be a subregion of the conformal boundary of an Anti-de Sitter spacetime, let $\text{CFT}$ denote the associated conformal field theory, and let $\Sigma$ be the minimal surf…

Consider an ancient asymptotically cylindrical 4-dimensional Ricci flow, meaning a Ricci flow existing for all sufficiently negative times and asymptotic at negative infinity to a…

Thurston's conjecture. This theorem should have a simpler proof based on more general principles, in particular the max flow/min cut principle, convexity, and…

Grayson's conjecture. There exists an example of curve shortening flow whose limit geodesics are not unique.

Global-maximality conjecture. For $n$, for only slightly larger values of $t$ than $\frac{3}{n+2}$, this local maximum is a global maximum.

Hamilton's conjecture. At least in dimension four, every linearly stable $GRS^+$ metric is an Einstein metric with positive scalar curvature.