90 problems

Conjecture 1 (Generalized Chang--Yang conjecture). For every integer $N\geq 1$ and every $\alpha\geq\frac{1}{2}$, let

Pointwise positivity conjecture. The estimates

Schoen's Weyl-vanishing conjecture. If $\overline{y}$ is a blow-up point, then the Weyl tensor $W_{g_0}$ of $g_0$ vanishes at $\overline{y}$ to order $\left[\frac{n-6}{2}\right]$;…

Let the DSII hierarchy be the integrable hierarchy governing deformations of surfaces immersed in $\mathbb{R}^4$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$ via generalized Weierst…

Let $n\geq 3$, let $(M^n,g)$ be a smooth compact Riemannian manifold with nonempty boundary, and suppose that $\lambda_1(L_g)>0$. For a conformal metric $\widetilde g\in[g]$ with z…

Let $(M,\mathcal{C})$ be a transverse Riemann-Lorentz conformal manifold of dimension $m\geq 4$, with singular set $\Sigma$. Around each singular point $p\in\Sigma$, consider a coo…

Let $(M,\mathcal{C})$ be a Riemann-Lorentz conformal space with polar end $D^{\infty}$. Suppose that $D^{\infty}$ is conformal-flat. A simultaneity distribution is a distribution w…

Let $n\leq 16$, and let $\mathcal{O}$ be an elliptic $n$-orbifold. Write $V_{c}(\mathcal{O})$ for its conformal volume. Uniform conformal-volume conjecture. There is a function…

Let $\Sigma$ be a closed surface, and let $I(\Sigma)$ denote its Friedlander–Nadirashvili invariant, defined as the infimum over conformal classes $C$ on $\Sigma$ of the supremum…

Nonpositive-curvature local diffeomorphism conjecture. If

Let $(M,g)$ be a complete, noncompact manifold with only one end: outside a compact set it is diffeomorphic to a spherical shell $S^{n-1}\times\mathbb{R}$. Let $g_0$ denote the Euc…

Let $K$ be a knot, and let $|csl|(K)$ denote the absolute value of its conformal linking energy. Positive lower-bound conjecture. There is a positive constant $C$ such that, if $K$…

Let $K$ be a knot and let $\psi_K^{(2,2)}$ be its twice tangent sphere map, which assigns to a pair of distinct points of $K$ the sphere tangent to $K$ at both points, wherever thi…

Let $W$ denote the conformal energy of a simplicial Willmore sphere. Quantization conjecture. The conformal energy is quantized: … This conjecture is motivated by numerical experim…

Meeks–Sullivan conjecture. The surface $M$ is parabolic.

Let $M^n$ be a closed connected smooth $n$-manifold whose Stiefel–Whitney classes all vanish. A Riemannian metric is conformally flat when its Weyl tensor vanishes. Conformal-flat…

Compactness and existence conjecture. If $A_g\in\Gamma_k^+$, then there exists a conformal deformation $\tilde{g}=e^{2u}g$ such that

Let $M$ be a topological $2$-sphere, and let $g_i=e^{2u_i}g_0$ be a family of conformal metrics with bounded area and energy. Assume … Suppose there is exactly one bubble point…

Conformal concordance conjecture. If $C_0$ and $C_1$ are conformally concordant, then they are isotopic in ${\cal C}^+(M)$.

In the Bach case, the weights satisfy … The Bach action is the squared norm of the Weyl curvature, and the problem is varied independently with respect to the metric and connection…

Let $(X^N,g_+)$, with $N=n+1\ge 3$, be a smooth Poincaré–Einstein manifold with conformal infinity $(M^n,[h])$. Assume that $Y(M,[h])>0$, and let $(\overline X,[\bar g])$ denote it…

Let $(X,g_+)$ be a conformally compact Einstein manifold with compactification $(\overline X,\overline g)$ and boundary $N$, and suppose that $\Lambda^\gamma(N,[\overline g|_N])>0$…

Let $\mathcal{R}$ and $\mathcal{R}'$ be two surfaces in $\mathcal{S}$. Suppose the cell of $\mathcal{R}$ at position $(i,j)$ has cross ratio $k_{i,j}$, while the cell of…

Let $k,m\in\mathbb{N}$ and let $N(k,m)$ be the dimension threshold appearing in the higher-order GJMS geometric Aubin set conjecture for conformal manifolds virtually conformal to…

The higher-order GJMS geometric Aubin set conjecture. The set $\mathscr{A}_{n,2k}$ is a geometric Aubin set for $L_{2k}$.