Let $\Sigma \subset \mathbf{S}^{n}$ be a codimension one, embedded minimal surface in the round $n$-dimensional sphere. Let moreover $\mathbf{C} = \mathbf{C}(\Sigma)$ be the minimal cone in $\mathbf{R}^{n+1}$ whose link is $\Sigma$. This is called a homology sphere if it has the same homology as $\mathbf{S}^{n-1}$, that is $H_0(\Sigma,\mathbf{Z}) = \mathbf{Z} = H_{n-1}(\Sigma,\mathbf{Z})$ and $H_i(\Sigma,\mathbf{Z}) = 0$ for all other $i$.
- The topology of this link---in the sense of whether or not is a homology sphere---is relevant in some constructions. For example White [1] constructs complete minimal surfaces asymptotic to a given area-minimising $\mathbf{C}$ if its link $\Sigma$ is not a homology sphere (along with some other hypotheses). White points out that all area-minimising cones (known at the time) had links that were not homology spheres.
- On the other hand, in a series of papers Hsiang constructed examples of non-equatorial minimal hyperspheres in $\mathbf{S}^n$. Although I may be misinterpreting Hsiang's results, as far as I understand none of the corresponding cones are known to be area-minimising, but Hsiang and Sterling [2] proved that some of them are stable, for a range of dimensions that is large---roughly $n \geq 20$ if I am not mistaken---and conjectured unbounded.
Question 1. Does there exist an example of a homology sphere $\Sigma^{n-1}$ minimally embedded in $\mathbf{S}^n$ so that $\mathbf{C} = \mathbf{C}(\Sigma)$ is area-minimising? For instance is any of the examples of Hsiang--Sterling known (or suspected) to be minimising?
Question 2. In the other direction, is there an example of a homology sphere $\Sigma$ so that $\mathbf{C}(\Sigma)$ is stable but not area-minimising? Morally speaking, should this be the rule or the exception?
[1] White. New applications of mapping degrees to minimal surface theory. J. Differential Geom. 29 (1989). no. 1, 143-162.
[2] Hsiang and Sterling. On the construction of nonequatorial minimal hyperspheres in $S^n(1)$ with stable cones in $\mathbf{R}^{n+1}$. Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 24, Phys. Sci., 8035-8036.
