9 problems

Let $M^n$ be an embedded closed minimal hypersurface in the unit sphere $S^{n+1}(1)$, and let $\lambda_1(M)$ denote its first Laplace--Beltrami eigenvalue. Yau's conjecture. … Yau'…

Let $M$ be a closed simply connected manifold admitting a Riemannian metric of nonnegative sectional curvature. Grove's conjecture. Then $M$ has a DDBD structure, that is, an isopa…

Let $M$ be an isoparametric hypersurface or a focal submanifold of the unit sphere $\mathbb{S}^n$. Tang's conjecture. Every such isoparametric hypersurface and focal submanifold ad…

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…

Let $\Omega\subseteq\mathbb{S}^{n+1}$ be a domain with smooth connected boundary. Suppose there is a positive function $u\in C^2(\overline{\Omega})$ and a function…

Let $(N^{n+1},\mathbf{g})$ be a compact Riemannian manifold with boundary $\partial N$, and let $(M_c,\bar{\mathbf{g}}|_{M_c})$ be an isoparametric hypersurface in the unit sphere…

Let $M_+$ and $M_-$ be the focal submanifolds of an isoparametric family with $(g,m)=(6,2)$, viewed as submanifolds of $\mathbb{R}^{14}$. A cubic form is a homogeneous polynomial o…

Let $\Omega$ be a harmonic domain in the unit sphere ${\bf S}^{n}$. A domain is an isoparametric tube if it is a solid tube of constant radius around a closed, smooth, minimal subm…

Let $\Omega$ be a Euclidean domain, and let $\lambda$ be a fixed Dirichlet eigenvalue of $\Omega$. A solution of the overdetermined problem … for a constant $c$ should force…