7 problems

Consider the Carlotto–Schulz minimal hypersurfaces with $k=\ell>1$. Their stability index is at least $k^2+6k+8$. Stability-index formula conjecture. The stability index of these h…

Let $M etext{a totally geodesic hypersurface}$ be a minimal hypersurface in $mathbb S^{n+1}$, and let $operatorname{ind}(M)$ denote the number of negative eigenvalues, counted with…

Index characterization conjecture. The Clifford hypersurfaces are the only minimal hypersurfaces with stability index $N+3$. This conjecture is known when $N=2$ by Urbano, but the…

Let $M$ be a compact minimal hypersurface, and let $J$ be its stability operator. The stability index of $M$ is the number of negative eigenvalues of $J$, counted with multipliciti…

Let $\mathcal{M}^n\subset \mathbb{S}^{n+1}$ be a compact minimal hypersurface that is not an Euclidean sphere, and let $\mathrm{ind}(\mathcal{M}^n)$ denote the number of negative e…

Index conjecture. Numerical experimentation as $n\rightarrow\infty$ suggests that

Stability-index conjecture. The only minimal hypersurfaces in $S^{n+1}$ with stability index $2n+5$ are the isoparametric hypersurfaces with two principal curvatures.