10 problems

Cone hot spots conjecture. A statement similar to the paper's theorem for cones should hold for these manifolds with boundary.

Weighted Riemannian warped-product curvature conjecture. The $N$-warped product of $B$, $F$, and $f$ satisfies $\mathsf{CD}(K(N+d-1),N+d)$ if $(B,f^{N+d}\operatorname{vol}_B)$ is a…

Let $g=dr^2+\varphi(r)^2h$ be a log-concave warped product metric, where $h$ is a flat metric on a torus or the standard metric on a sphere, and suppose that the warped product spa…

Let $N$ denote the synthetic dimension parameter in the dimension bound for the RCD condition discussed above. Dimension-bound conjecture. The dimension bound $N+1$ can be improved…

Lorentzian warped-product timelike curvature-dimension conjecture. If $f”-Kf\leq0$ and $(F,d_F,\operatorname{m}_F)$ is non-branching and satisfies $\mathsf{CD}(K_F(N-1),N)$, then t…

Lorentzian warped-product curvature conjecture. The Lorentzian warped product $-I\times_fF$ has timelike CBB by $K$ if and only if $f$ is $-Kf$-concave and $F$ has CBB by $K_F$. Th…

Noncollapsed RCD warped-product conjecture. The $N$-warped product of $B$, $F$, and $f$ satisfies $\mathsf{CD}(K(N+d-1),N+d)$ if $B^\dagger$ is a noncollapsed…

Let $B$ be a metric-measure base of dimension $d$, let $F$ be an essentially non-branching metric-measure space, and let $f$ be a Lipschitz nonnegative warping function. Let…

Let $B\times_f^{N}F$ be an $N$-warped product of a Riemannian manifold $B$, a positive warping function $f$, and a weighted Riemannian manifold $F$. The weighted semi-Riemannian an…

Barrier conjecture. Rotationally symmetric graphs can be used as barriers to obtain sharp height estimates in general warped spaces $P\times_{e^{-\psi}}\mathbb{R}$ when the radial…