53 problems

Universal-cover conjecture. The universal covering space $\widetilde{M}$ is isometric to $G/\operatorname{\textsl{SU}}(2)$ equipped with its unique $G_2$ structure with parallel to…

Generalized Calabi–Markus conjecture. The fundamental group $\pi_1(M)$ is finite, and $M$ is noncompact.

Let $M^n\subset\mathbb{R}^{n+1}$ be a connected, complete, embedded self-shrinker with polynomial volume growth, and let $A$ be its second fundamental form. Sharp Gaussian curvatur…

Let $X$ be a compact Kähler manifold, and let $TX$ denote its holomorphic tangent bundle. A Hermitian metric $c9$ on $TX$ is assumed to be uniformly RC-quasi-positive, or equivalen…

Let $X^n$ be a closed manifold and let $f:X^n\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal no…

Let $X^4$ be a closed manifold and let $f:X^4\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal no…

Let $k\geq 2$. Let $S_1,\ldots,S_k\subseteq\mathbb{R}^{d}$ be smooth hypersurfaces equipped with measures $\sigma_1,\ldots,\sigma_k$, respectively. They are transversal if there is…

Let $(M^n,g)$ be a complete noncompact Kähler manifold with nonnegative curvature and maximal volume growth. Let $g(t)$, $t\in[0,+\infty)$, be the unique complete solution of the K…

Let $(M^n,g)$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature, and assume that the universal cover of $M$ does not split. For a point $p$, let…

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. For a $(1,0)$-vector $X$, let $\mathcal{C}_{\alpha,\beta}(X)$ denote the mixed curvature, formed from the relevant Ch…

Symmetrized curvature conjecture. If $1\leq k\leq \left[\frac{n}{2}\right]$ and $M$ has $k$-positive symmetrized curvature operator $\mathcal R$, then $M$ is biholomorphic to…

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. Let $\alpha$ and $\beta$ be fixed real numbers with $\beta\neq0$, and let $\mathcal{C}_{\alpha,\beta}$ denote the mix…

Let $\{M_n\}_{n\geq1}$ be a sequence of infinite random planar maps believed to lie in the universality class of $\gamma$-LQG, such as uniform infinite triangulations when…

Let $M$ be a Hermitian manifold with a Hermitian metric $g$. Assume that $g$ has positive or quasi-positive real bisectional curvature. Yang–Zheng's conjecture. Then $M$ is project…

Let $\mathcal M(n,r,V)$ be the class of $n$-dimensional complete Riemannian manifolds with sectional curvature at most $1$, injectivity radius at least $r>0$, and volume at most…

Let $X\subseteq\mathbb{P}_\mathbb{C}^3$ be a possibly singular K3 surface, and let $X_s$ be its smooth locus with curvature form $J$ induced by the Fubini–Study metric on…

Isolation conjecture. The hypersurface $X$ has isolated complex critical curvature points.

Let $M$ be an $n$-dimensional Kähler manifold with non-positive holomorphic bisectional curvature, and suppose that its Ricci curvature is quasi-negative. Li-Zheng's conjecture. Th…

Let $(M,g)$ be a closed, simply connected Riemannian manifold. Let $d40$ denote the curvature operator of the second kind, restricted as a bilinear form to the trace-free symmetric…

Let $\Gamma$ be a generic plane curve evolving under the curve-shortening flow, and let $\kappa$ denote its signed curvature. Write $n_\kappa(t)$ for the number of critical points…

Let $a<b$ and let $\epsilon$ satisfy … Consider the surface … An umbilic point is a point where the principal curvatures coincide, and its index is the index of the principal direc…

Let $a,b,c>0$ and let $k\in\mathds{Z}_{>1}$. Consider the surface … An umbilic point is a point where the principal curvatures coincide, and its index is the index of the principal…

Let $(M,\omega)$ be an $n$-dimensional compact Kähler manifold with nonpositive holomorphic bisectional curvature, and suppose its Ricci curvature is quasi-negative, meaning nonpos…

Let $S_i\subset\mathbb{R}^n$, $i=1,\ldots,k$, be smooth, compact hypersurfaces, where $1\leq k\leq n$. For functions $f_i$ supported in neighborhoods $U_i$ of $S_i$, write…

Let $(N^n,\mathfrak{F})$ be an isoparametric foliation. Suppose its leaves have nonnegative or positive leafwise sectional curvature, $K_{\mathrm{sec}}^F\geqslant0$ or…