704 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 $M$ be a compact immersed constant mean curvature surface without boundary in $\mathbb{R}^3$. Hopf's conjecture. The surface $M$ is the round sphere. The conjecture is a classi…

Chern's conjecture. The Euler characteristic of $M$ is zero.

Let $\xi_{g,1}\subset S^3$ be the Lawson minimal surface of genus $g$ indexed by $(g,1)$, and consider its stereographic projection to $\mathbb{R}^3$. Kusner's conjecture. For surf…

Let $M$ be a spin manifold with a positive Ricci curvature metric, and let its Witten genus be viewed as an invariant in the appropriate ring of modular forms. Stolz's conjecture.…

DDVV conjecture. The normal scalar curvature conjecture asserts that the normal scalar curvature of $M$ satisfies the DDVV inequality.

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…

Let $M$ be a complete embedded $δ$-stable minimal surface in a complete flat three-manifold, where $δ$-stability means that … for every $\varphi\in C_0^\infty(M)$. Meeks–Pérez–Ros…

Let ${\mathcal W}$ be a non-parallelizable $3$-web on a real or complex $2$-dimensional manifold. A $3$-web is linearizable if it is equivalent, up to a local diffeomorphism, to a…

Let $S$ be a complete minimal surface embedded in $\mathbb{R}^3$ and of finite total curvature. If $S$ is neither a plane nor a catenoid, let $n$ be its number of ends and let…

Let $X$ be a PA-space, let $f$ be a PA-form on $X$, let $\Omega(X)$ denote the space of smooth differential forms on $X$, and let $P$ be a differential operator. Differential-opera…

Let $\Sigma^n$ be a smooth closed hypersurface minimally embedded in the unit sphere $\mathbb{S}^{n+1}$ equipped with the round metric, and let $\lambda_1(\Sigma)$ denote the first…

Let $\mathbb S^3$ be the 3-sphere, and let a rotational torus mean a torus invariant under a rotational symmetry of $\mathbb S^3$. Pinkall–Sterling conjecture. The rotational torus…

A four-dimensional κ-noncollapsed steady gradient Ricci soliton is a complete steady gradient Ricci soliton satisfying the stated curvature conditions: it has nonnegative curvature…

Let $M$ be a complete biharmonic submanifold of Euclidean space. Akutagawa–Maeta conjecture. $M$ is minimal. This is a global reformulation of Chen's conjecture for complete subman…

Let $n\geq 3$. A minimal hypersurface is a hypersurface in Euclidean space $\mathbb{R}^n$ whose mean curvature vanishes. A minimal hypersurface is complete when its induced Riemann…

Let a convex smooth surface be a smooth convex surface in $\mathbb{R}^3$, and let a closed simple geodesic be a closed geodesic that is non-self-intersecting. Poincaré's conjecture…

Let $M$ be a compact nilmanifold and let $g$ be an Einstein metric on $M$. The Einstein metric conjecture. The metric $g$ is flat, and $M$ is a torus. This concerns the possible Ei…

Let $M$ be a complete embedded minimal surface in Euclidean space $\mathbb{R}^3$. Calabi–Yau conjecture. Every such surface is necessarily proper. The conjecture concerns the relat…

Let $(M,g)$ be an $n$-dimensional Cartan-Hadamard manifold with sectional curvature bounded above by $-\kappa^2$ for some $\kappa\geq0$. Let $N^n_\kappa$ be the corresponding model…

Vershik's conjecture. Despite examples in which such tensor fields have been defined and computed, there is probably no general definition of a curvature tensor for non-holonomic d…

Burghelea–Haller conjecture. The equality

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper stat…

A closed smooth curve is a smooth embedded curve in three-dimensional space. Steinhaus's conjecture. Every closed smooth curve in three-dimensional space has a pair of parallel tan…

Sasakian–Einstein conjecture. Every such homotopy sphere admits a Sasakian–Einstein metric.