154 problems

Maeta's generalized Chen conjecture. Any $k$-harmonic submanifold in the Euclidean space $\mathbb R^{n+1}$ is minimal.

The universal Teichmüller space is the space of boundary values of quasiconformal self-maps of the disk; equivalently, it can be viewed through quasisymmetric self-maps of…

Let $\Omega$ be a bounded open domain in an $n$-dimensional Alexandrov space with curvature bounded from below by $k$, and let $N$ be a compact smooth Riemannian manifold. Suppose…

Let $f:M\rightarrow N$ be a homotopy equivalence between closed negatively curved manifolds. The unique harmonic map $h$ homotopic to $f$ is thereby defined. Lawson–Yau conjecture.…

A map $f:X\to Y$ between metric spaces is a quasi-isometric embedding if it is coarse Lipschitz and, for some $M$, satisfies … The hyperbolic plane is denoted by \H^2, and two ma…

Normal-index-one conjecture. The normal index of $\phi$ is always equal to one:

Rivière's conjecture. There exists $\delta=\delta(C,n,N)>0$ such that

Hitchin WKB conjecture. As $t\to\infty$, for every non-critical path $\gamma$,

Let $M$ be a rank one symmetric space, and let a quasiconformal self-homeomorphism of the boundary at infinity of $M$ be a quasiconformal homeomorphism from that boundary to itself…

Higher-order harmonicity conjecture. The function $G(r)$ is proper $(k+1)$-harmonic for every $k\geq 1$.

Bizoń et al.'s blow-up profile conjecture. The rescaled solution converges to the static profile:

Let $f:S^1\to S^1$ be a quasi-symmetric selfmapping of the unit circle. A map of the unit disk is hyperbolic-harmonic if it is harmonic for the hyperbolic metric, and it is quasico…

Let $S$ and $R$ be Riemann surfaces, let $f_1:S\to R$ be a quasiconformal map, and let $\mathcal{CM}(R)$ be the space of conformal metrics on $R$ with unit area and at most conical…

Let $k\leq m$, let $n\in\mathbb{N}$, and let $N^k$ be a $k$-dimensional closed submanifold of $\mathbb{R}^m$. For every $C>0$, consider $W^{1,2}$ harmonic maps $u:B^n_2(0)\to N^k$…

The quaternionic and octonionic projective spaces in question are … A second elliptic problem is the second elliptic system associated to the corresponding 4-symmetric space. Geome…

A 4-symmetric space is a homogeneous space equipped with an automorphism of order four, and a second elliptic system is the integrable system associated to such a space. Geometric…

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let…

Direction-of-approach conjecture. The surface parts of the limit image should depend only on the boundary point of moduli space approached, whereas the complex of straight line seg…

Uniqueness conjecture. It may be possible that there is only one boundary point of the compactified moduli space having a finite-energy limit of map energies for the given set of m…

For every integer $m>10$, there is a harmonic cellular map $h:M_1\rightarrow M_2$ between a pair of closed negatively curved $m$-dimensional Riemannian manifolds that is not a diff…

Hitchin–Goldman conjecture. The function $e$ has a unique minimum.

Let $M$ and $N$ be closed negatively curved Riemannian manifolds, and let $h:M\to N$ be a harmonic map that is not a diffeomorphism but is the limit of a 1-parameter family of diff…

Let $\mathsf{Harm}_E(S^{2m})$ denote the space of harmonic maps $S^2\to S^{2m}$ having energy $4\pi E$. Bolton–Woodward's conjecture. The space $\mathsf{Harm}_E(S^{2m})$ is a compl…

Let $n\geq 2$, and consider the space of harmonic maps $S^2\longrightarrow\mathbb{CP}^n$ with fixed energy and fixed degree. Connectedness conjecture. This space is connected. The…

Verdier's conjecture. The space $\mathhs$ is path connected.