13 problems

Let $\Sigma_a$ be the critical spherical catenoid with parameter $a>1/2$. Decompose its Robin Morse index into Fourier modes $|k|=0$, $|k|=1$, and $|k|\geq 2$. Mode-by-mode index c…

Let $(N^{n+1},g)$ be a closed Riemannian manifold with positive Ricci curvature. A closed embedded minimal hypersurface is a smooth hypersurface $M^n\subset N^{n+1}$ with zero mean…

Let $\gamma\in\mathbb{N}$, and let $\Sigma\subset\mathbb{S}^3$ be a closed embedded minimal surface of genus $\gamma$. Denote its Morse index by $\operatorname{Ind}(\Sigma)$, and l…

Let $x:M^n\rightarrow\mathbb{S}^{n+1}$ be a full, closed, minimal hypersurface in $\mathbb{S}^{n+1}$, with $n\geq 3$. Here $\operatorname{Ind}(M)$ denotes the Morse index of $M$. P…

Let $M$ be a closed minimal surface in $\mathbb{S}^3$ of genus $\gamma$, and let $\operatorname{ind}(M)$ denote its Morse index. Let $\xi_{\gamma,1}$ be the Lawson surface, whose i…

Let $b\geq3$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Maximal-are…

Let $n\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let…

Let $(M,g)$ be an ambient manifold with positive Ricci curvature, and let $\Sigma$ be an embedded orientable compact minimal hypersurface of index $I$. Marques–Neves' Betti-number…

Index-one Clifford hypersurface classification conjecture. The only compact minimal hypersurfaces with index one in $\mathbb{K}P^{r-1}$ are the minimal Clifford hypersurfaces. In p…

Let $M$ be a compact oriented minimal surface in a flat three-torus, and suppose that $M$ has genus three. Write $index_A$ for its Morse index with respect to the area functional.…

Let $M^3$ be a Riemannian $3$-manifold, and let a min-max limit arising from a Heegaard splitting have components $\Gamma_i$. Call a component unstable if it is unstable as a minim…