Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\operatorname{dist}(T,0) = 1$.

For all $R > 1$, $T$ intersects $\partial B_R$ transversely. Thus $T \cap \partial B_R$ is a smoothly embedded submanifold of $\partial B_R$; rescale this homothetically to $\Sigma_R \subset \partial B_1$.

• When $R$ is close to $1$, $\Sigma_R$ is diffeomorphic to the boundary of an $2n-1$-dimensional disk, so $\Sigma_R \simeq \mathbf{S}^{2n-2}$.
• As $R \to \infty$, $\frac{1}{R} T \to \mathbf{C}_S$, so when $R$ is large enough then $\Sigma_R \simeq \mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$.

Question. The paradox is that the family $(\Sigma_R)$ gives an isotopy from $\mathbf{S}^{2n-2}$ to $\mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$, where we let $R$ vary across $[1+\frac{1}{N},N]$ for example. What am I missing here?

Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\operatorname{dist}(T,0) = 1$.

For all $R > 1$, $T$ intersects $\partial B_R$ transversely. Thus $T \cap \partial B_R$ is a smoothly embedded submanifold of $\partial B_R$; rescale this homothetically to $\Sigma_R \subset \partial B_1$.

• When $R$ is close to $1$, $\Sigma_R$ is diffeomorphic to the boundary of an $2n-1$-dimensional disk, so $\Sigma_R \simeq \mathbf{S}^{2n-2}$.
• As $R \to \infty$, $\frac{1}{R} T \to \mathbf{C}_S$, so when $R$ is large enough then $\Sigma_R \simeq \mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$.

Question. The paradox is that the family $(\Sigma_R)$ gives an isotopy from $\mathbf{S}^{2n-2}$ to $\mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$, where we let $R$ vary across $[1+\frac{1}{N},N]$ for example. What am I missing here?

Edit. As pointed out in the answers below, the first point is incorrect. I mixed up two arguments I thought up: the first working with balls centered around a point of the surface—this guarantees the validity of both bullet points—, and the second working with balls centered at the origin—guaranteeing the transversality. Presumably in the former approach the transversality would be false for some radii, which would allow the topology to change.

Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be the leaf of the Hardt–Simon foliation with $\operatorname{dist}(T,0) = 1$.

For all $R > 1$, the support of $T$ intersects $\partial B_R$ transversely. Thus $\operatorname{spt} \lVert T \rVert \cap \partial B_R$ is a smoothly embedded submanifold of $\partial B_R$; rescale this homothetically to $\Sigma_R \subset \partial B_1$.

• For $R$ close enough one to $1$, $\Sigma_R$ is diffeomorphic to the boundary of an $2n-1$-dimensional disk, so $\Sigma_R \simeq \mathbf{S}^{2n-2}$.
• As $R \to \infty$, $\frac{1}{R} T \to \mathbf{C}_S$, so when $R$ is large enough then $\Sigma_R \simeq \mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$.

Question. It would seem that this allows the absurd conclusion that the family of surfaces $(\Sigma_R)$ with $R \in [1+\frac{1}{N},N]$ say, gives an isotopy from $\mathbf{S}^{2n-2}$ to $\mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$. What am I missing here?

Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\operatorname{dist}(T,0) = 1$.

For all $R > 1$, $T$ intersects $\partial B_R$ transversely. Thus $T \cap \partial B_R$ is a smoothly embedded submanifold of $\partial B_R$; rescale this homothetically to $\Sigma_R \subset \partial B_1$.

• When $R$ is close to $1$, $\Sigma_R$ is diffeomorphic to the boundary of an $2n-1$-dimensional disk, so $\Sigma_R \simeq \mathbf{S}^{2n-2}$.
• As $R \to \infty$, $\frac{1}{R} T \to \mathbf{C}_S$, so when $R$ is large enough then $\Sigma_R \simeq \mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$.

Question. The paradox is that the family $(\Sigma_R)$ gives an isotopy from $\mathbf{S}^{2n-2}$ to $\mathbf{S}^{n-1} \times \mathbf{S}^{n-1}$, where we let $R$ vary across $[1+\frac{1}{N},N]$ for example. What am I missing here?