232 problems

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 $M$ be a compact Riemannian manifold, and let $K$ be a curvature formed locally from the Riemannian curvature tensor, its covariant derivatives, and the metric. A conformal inv…

Lorentzian Lichnerowicz conjecture for conformal vector fields. If $(M,g)$ is not conformally flat, then every conformal vector field on $M$ is inessential.

Sullivan's and Schoen–Yau's conjectures. Every proper minimal surface in $\mathbb{R}^3$ having finite topology, as conjectured by Sullivan, or having a proper projection into a pla…

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

Let $(\overline{X},\bar{g})$ be a smooth compact Riemannian manifold with boundary, and let $\Theta_n(\overline{X},[\bar{g}])$ denote the supremum of the isoperimetric ratio over s…

Schoen's conjecture. At a blow-up point, the Weyl tensor should vanish up to $[\frac{n-6}{2}]$-th order derivatives.

In conformal geometry, let $(M,g)$ be a smooth, compact, aspherical Riemannian manifold, and consider the set of Yamabe metrics on $M$. Compactness Conjecture. This set is compact.…

Let a Lorentzian conformal manifold be a manifold equipped with a conformal class of Lorentzian metrics, and let $\operatorname{Ein}^{1,n}$ denote the Lorentzian Einstein space. Tw…

Belgun–Moroianu's conjecture. Every closed Weyl structure on a compact manifold is flat, exact, or has irreducible holonomy.

Let $T(a,b)=\mathbb{R}^2/\Gamma(a,b)$ be the flat torus associated with the lattice $\Gamma(a,b)=\mathbb{Z}(1,0)\oplus\mathbb{Z}(a,b)$, where $(a,b)$ lies in the moduli region … Wr…

Hang–Yang conjecture. Every such solution $v$ is constant.

Let $\Sigma^k$ be an oriented surface of topological genus $k\geq 1$, let $\iota:\Sigma^k\to\mathbb{S}^3$ be an immersion, and let $\mathcal{W}_{\iota}(\Sigma^k)$ denote its Willmo…

The conformal-invariant extension conjecture. Theorem t-rigidity and Corollary c-asf hold assuming only one of the conformal invariant conditions

Let $(M,g_0)$ be a smooth compact $n$-dimensional Riemannian manifold with boundary $\partial M$, where $n\geq 3$, and let $Y(M,\partial M,[g_0])$ denote its generalized Yamabe con…

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 $S^{4n+3}$ be the standard unit $(4n+3)$-dimensional quaternionic sphere with its standard quaternionic contact structure and standard 3-Sasaki structure. Its qc-scalar curvatu…

Consider the conformal class $[g_F]$ defined by the metric in equation, arising from the defining function … where $a_i$, $i=0,1,\ldots,6$, and $b$ are real constants. Let $H$ deno…

Let $n$ be a dimension and let $\mathcal{O}$ be an elliptic $n$-orbifold. Denote by $V_{PC}(\mathcal{O})$ its conformal volume in the sense used in the paper. Uniform conformal vol…

Let … For a positive function $u$ on $S^1$, let ${\bf F}(u)$ denote the functional whose minimizers are being considered. Extremal classification conjecture. If $u$ is a minimizer…

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…

Green-function coefficient conjecture. At this point,

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…

A natural density $U$ is a scalar density constructed naturally from the metric and its derivatives, and a Polyakov-type formula has a local part consisting of terms integrated aga…