5 problems

Yamabe product conjecture. The Yamabe constant $Y(g(T))$ is minimized by $u_{n,T}$, and

Let $M$ and $N$ be compact Riemannian manifolds. For a fixed volume fraction, consider the product of $N$ with a sufficiently small homothetic copy of $M$, equipped with the corres…

The Strichartz conjecture on $\mathbb{R}\times\mathbb{S}^3$. One has

Let $S^{d_1}\times S^{d_2}\times\cdots\times S^{d_n}$ be a product of spheres of length $n\geq 5$, let $d=d_1+\cdots+d_n$, and suppose the Weyl error bounds have the form…

Let $igma^*_ambda$ denote the constructed constant mean curvature pieces, with parameter $ambda$ and mean curvature $H$. In $^2\timesmathbb{R}$ and $^2\timesmathbb{R}$, consider th…