94 problems

Let $(M^n,g)$ be a smooth compact Riemannian manifold with nonempty boundary, of positive conformal type, and not conformally equivalent to the round hemisphere. Consider positive…

Let $n\geq 3$ and let $(M^n,g)$ be a compact Riemannian manifold with nonempty boundary and positive Yamabe–Escobar invariant. Consider the normalized set of conformal metrics…

For every closed Riemannian manifold $(M^n,g)$ with $n\ge 7$, if $Q_g^{(2)}>0$, $Q_g^{(4)}>0$, $Q_g^{(6)}>0$, and the sixth-order GJMS operator $P_{6,g}$ is strictly positive as a…

Let $\Gamma$ be a closed Jordan curve, let $h$ be its conformal welding, let $C_h$ be the pull-back operator on $\mathrm{BMO}$, and let $A_h$ be the analytic projection of $C_h$. D…

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

Let $M$ be a closed Lorentzian conformal manifold. It is essential if its conformal transformation group does not preserve any metric in the conformal class. Lorentzian Lichnerowic…

Green function rigidity conjecture. Then $(M,g)$ is a round sphere.

Hamilton's conjecture. The metric $g(t)$ converges to a metric of constant scalar curvature as $t\to\infty$.

Yamabe's conjecture. There exists a metric $g$ on $X$ conformally equivalent to $g_0$ and having constant scalar curvature.

Let $M^n$ be a Riemannian manifold with $n$ even, and let $P(g)$ be a Riemannian invariant of weight $-n$ such that … is unchanged under conformal rescalings $g\mapsto e^{2\phi}g$…

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…