348 problems

The conjecture asks for the sharp Sobolev trace inequality associated with $\sigma_2$-curvature on the round hemisphere, together with a classification of its equality cases, equiv…

For every integer $n\ge 3$ and every convex planar $n$-gon $P$ of area $A$, let $\lambda_1^N(P)$ denote the first nonzero eigenvalue of the Laplace operator with Neumann boundary c…

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…

Let $(M,I,J,K,g_0)$ be a compact HKT manifold, let $\Omega_0$ be the HKT form of the initial metric, and let $T_{\max}$ denote the maximal existence time of the unnormalized HKT-Ri…

Let $Y$ be a projective Calabi–Yau variety with only ordinary double points, and let $X\to Y$ be a crepant resolution or let $Y$ occur as the central fiber of a smoothing. Then the…

Let $S$ be a horizontal strip in $\mathbb{R}^2$, and let the density be $\rho(x)=1$ on $S$ and $\rho(x)=\lambda>1$ on $\mathbb{R}^2\setminus S$. For every $\lambda>1$ and every adm…

Let $X$ be a compact Riemann surface and let $(E,\Phi)$ be a stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundle on $X$. For each $r>0$, let $e_r(x)$ denote the energy density at…

Every connected minimal hypersurface $M^3\subset S^4$ with constant scalar curvature is isoparametric; equivalently in the local formulation, $M$ is locally congruent to an isopara…

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $1<p<\infty$, and let $u\in C(\Omega)\cap W^{1,p}(\Omega)$ be continuous and Lebesgue monotone. Determine whether $u$ can…

For every compact set $K\subset\mathbb{R}^{2}$, let $\operatorname{cap}_{\log}(K)$ denote its logarithmic capacity and let $\operatorname{Cap}_{N}(K)$ denote the Newtonian (electro…

For every integer $n\ge 3$, every $\varphi\in L^p(\mathbb{S}^n)$ with $\frac{n}{n-1}<p<\infty$, and every vector field $A$ such that $\mathrm{d}A^\flat\in L^{n/2}(\mathbb{S}^n)$ an…

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain, let $\omega\in C^2(\Omega)\cap C(\overline{\Omega})$ be positive on $\overline{\Omega}$, and equip…

Let $T_{\mathrm{Cl}}\subset\mathbb{C}P^2$ denote the Clifford torus, and for a closed immersed surface $\Sigma\subset\mathbb{C}P^2$ define…

For every closed smooth strong spacelike curve $\gamma_0\subset\mathbb{R}^{2,q}$ of index $1$, the solution $\gamma(\cdot,t)$ of the usual curve-shortening flow exists up to a fini…

For the hyperbolic Hénon-type equation with exponential nonlinearity studied by Huang and Zhao, is every radial solution stable whenever $n\ge 10+4\alpha$ (with $\alpha>-2$)?

For every admissible Bartnik boundary datum $(\Sigma,\gamma,H)$, the Bartnik mass…

For each pair of dimensions $n,m$, there exists $\varepsilon_{n,m}>0$ such that every compact minimal submanifold $M^n\subset S^{n+m}$ satisfying $M\cong S^n$ and…

Let $C\subset\mathbb{S}^3$ be a great circle. Is every embedded non-orientable minimal surface $\Sigma\subset\mathbb{S}^3$ with boundary $\partial\Sigma=C$ congruent to the Lawson…

For every metric-measure space $(X,d,\mathfrak m)$, every $K\in\mathbb R$, and every $1<N<\infty$, is $\mathrm{BCD}(K,N)$ equivalent to $\mathrm{RCD}(K,N)$? Equivalently, does the…

For every compact Kähler surface $M$ and every maximal solution $g(t)$ of the Kähler–Ricci flow $\frac{\partial}{\partial t}g(t)=-\operatorname{Ric}(g(t))$ on $[0,T)$ with…

Every smooth properly embedded minimal surface $\Sigma\subset \overline{\mathbb{B}}^{3}$ that is topologically an annulus, satisfies $\partial\Sigma\subset \mathbb{S}^{2}$, and mee…

Let $G$ be the $15$-dimensional octonionic Heisenberg group. Every nontrivial entire finite-energy positive solution $u$ of the Yamabe-type equation on $G$ is, up to a left transla…

Let $X$ be a compact Kähler manifold, and let $\omega(t)$ solve the unnormalized Kähler–Ricci flow $\partial_t\omega(t)=-\operatorname{Ric}(\omega(t))$ on $[0,T)$, where $T<\infty$…

For every admissible compact initial-data set $(\Omega,g,k)$ with boundary $\Sigma$ for which the Wang--Yau quasi-local mass $\mathcal{M}_{\mathrm{WY}}(\Sigma)$ is defined and the…

For every integer $n\ge 2$, every complete simply connected Riemannian $n$-manifold $(M^n,g)$ with sectional curvature $\operatorname{Sec}_g\le \bar k\le 0$, and every volume $V>0$…