24 problems

Let $M$ be an irreducible compact Kähler manifold with positive orthogonal bisectional curvature. Projective-space conjecture. Is $M$ necessarily biholomorphic to…

The existence conjecture. Every simply connected Riemannian manifold $(M,\mathtt{g})$ with positive (respectively, non-negative) sectional curvature admits a codimension-one singul…

Let $(M^n,g_0)$ be a complete WPIC1 manifold with $n\geq 3$. WPIC1 Ricci-flow existence conjecture. There exists a complete WPIC1 Ricci flow $g(t)$ on $M$ for $t\in[0,T)$, for some…

Let $M$ be an open, complete, noncompact manifold without boundary whose curvature is PIC1. PIC1 open-manifold topology conjecture. $M$ is diffeomorphic to Euclidean space. This co…

Let $\pi\colon (M^{n+k},\textsl g)\to (B^n,b)$ be a Riemannian submersion. Suppose that $(M,\textsl g)$ is compact and has positive sectional curvature. Petersen–Wilhelm's conjectu…

Let $F\hookrightarrow M\rightarrow B$ be a Riemannian submersion from a positively curved closed manifold $M$. Petersen–Wilhelm's fiber dimension conjecture. Then … This conjecture…

Conjectured estimate. Estimate (2) is well-defined and holds.

Conjecture on quasi-positive curvature. If $X$ has a Kähler metric with quasi-positive holomorphic sectional curvature, then $X$ is a projective and rationally connected manifold.

Yang's conjecture. For every integer $p$ with $1<p<n$, the bundle $\bigwedge^p T_X$ admits a smooth RC-positive metric. In particular, if $X$ is Kähler, then $X$ is a projective an…

Let $T^1$ act isometrically and effectively on $X^4$, where $X^4$ is a $4$-dimensional, closed, positively curved, orientable Alexandrov space. Classification conjecture. Up to equ…

Wilhelm's conjecture. Every such submersion satisfies

Let $M$ be a compact manifold with positive curvature, and let $\cal F$ be a Riemannian foliation on $M$. A horizontal vector is a vector tangent to the horizontal distribution of…

Let $(M^n,g)$ be a complete Riemannian manifold of positive sectional curvature. A manifold has maximum volume growth when its volume growth is maximal in the relevant comparison s…

Topological classification conjecture for $|\calO|$. The underlying space $|\calO|$ is homeomorphic to either

Let $M$ be a compact oriented $4$-manifold with positive sectional curvature and nonvanishing second Betti number. Its intersection form is the bilinear form on $H^2(M;\mathbb{R})$…

Let $X$ be an $n$-dimensional Riemannian manifold with sectional curvature $K\geq 1$, and let $f:X\to\mathbb{R}^k$ be a continuous map. For a subset $Y\subset X$, write…

Let $M^{2m-1}$ be a closed Riemannian manifold with positive sectional curvature. Let $q$ denote the smallest prime divisor of $m$, and let an $r$-torus act effectively and isometr…

Let $M$ be a positively curved $G$-manifold with principal isotropy group $H\ne\{1\}$. Write ${}^r\!M$ for the core resolution of $M$. Core-resolution conjecture. Either … or … In…

Let $\mathbb{S}^{2}\times\mathbb{S}^{2}$ be the product of two 2-spheres. Hopf's product conjecture. There is no Riemannian metric on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ with posi…

Let $M$ be a closed Riemannian manifold with positive sectional curvature. Consider the flow defined in the paper, denoted by … converges uniformly to a nonzero Killing vector fiel…

Let $M^n$ be a simply connected positively curved manifold, and let $\Lambda M$ denote its loop space. The Betti numbers of $\Lambda M$ are the homology ranks with coefficients in…

Let $M$ be a simply connected positively curved 4-manifold equipped with an isometric circle action. A linear circle action is a circle action induced by a linear action on either…

Let $M$ be an orientable closed positively curved $4$-manifold with an effective isometric $S^1$-action. Hsiang–Kleiner's conjecture. $M$ is diffeomorphic to $S^4$ or ${\bf CP}^2$.…

Let $M$ be an $m$-dimensional Riemannian manifold of positive sectional curvature. Let $N$ and $H$ be totally geodesic closed submanifolds of dimensions $n$ and $h$, respectively,…