123 problems

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…

A compact, simply connected manifold $M$ is called rationally elliptic when … It is called rationally hyperbolic otherwise. Bott–Grove–Halperin conjecture. Every compact simply con…

Let $X$ be a complete non-compact Kähler manifold of complex dimension $n$. For a Kähler metric, write $BK>0$ when its bisectional curvature is positive: for all real unit tangent…

Hamilton's conjecture. If $(M,g(0))$ is smooth, connected, complete and Ricci pinched, then it is either Riemannian flat or compact.

Yau's uniformization conjecture. Any complete Kähler metric with non-negative bisectional curvature is biholomorphic to $\mathbb{C}^n$.

The setting is a hypergraph constructed by uniformly sampling a Riemannian manifold in the natural way, with coarse scalar curvature and the manifold's scalar curvature compared as…

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…

Curvature-radii pinching conjecture. The body $K$ is necessarily a ball of radius $C$.

Let $M$ be a compact Riemannian manifold. Its curvature operator is positive when all of its eigenvalues are positive. Hamilton's conjecture. In every dimension, if $M$ has positiv…

Let $M$ be a strictly pseudoconvex CR manifold with contact form $\theta$, Tanaka–Webster connection $\nabla$, and associated metric $g_\theta$. Fix $x\in M$ and a $2$-plane…

Let $M$ be a Carnot–Carathéodory complete Sasakian manifold with pseudohermitian Ricci tensor $\rho$, and let $(\varphi,\xi,\eta,g)$ be its contact metric structure. For some…

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

Hedenmalm–Perdomo sharpness conjecture. The number $0.52$ can be replaced by $\frac12+\varepsilon$ for any positive $\varepsilon$ in the curvature condition, while retaining the co…

Let $\boldsymbol\mu(z)=\mu(z)\,d\Sigma(z)$ be the associated $2$-form of a $C^\infty$-smooth real-valued function … boldsymbolmu(z)+frac{alpha}{2},mathbf K{mathbb H}(z)le 0,qquad z…

Sectional curvature bounds conjecture. Both the non-positivity of the sectional curvatures of ${\cal K}_1$ and the lower bound $-\frac{1}{2}n(n-1)$ should hold for Calabi--Yau mani…

Volume-coefficient conjecture. If all coefficients $c_{2k}$ vanish, then $M$ is flat.

Curvature and flatness conjecture. The reasoning in (3.2.2) should extend to the general case so that the sectional curvature of the hermitian connection on…

Let an AdS Kerr black hole in $d$ dimensions be a higher-dimensional rotating black-hole solution with a negative cosmological constant, and let its Ricci-flat Kerr counterpart be…

Let $\mathfrak{M}_{n|reg}$ be the regular thermo-metric event manifold, with curvature $2$-form $\mathfrak{R}^{ij}$. At the origin, the curvature $2$-form has the same form as that…

For a 2-soft monohedral tiling of $\mathbb{R}^3$, let $K_E$ denote the total Gaussian curvature integrated along all edge strata. The Gaussian-curvature bound. We conjecture that ……

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 complete non-compact Kähler manifold of complex dimension $n$. Assume that $X$ admits a Kähler metric with positive sectional curvature, which is stronger than the pos…

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…

Harmonic mapping conjecture. Every stable harmonic map $f:M\to N$ must be constant.