Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth: $\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \rVert < \infty$. Let $\mathbf{C}_0$ be a tangent cone to $M$ at the origin, and $\mathbf{C}_\infty$ be a tangent cone ‘at infinity’, meaning obtained by blowing down $M$.

To narrow the problem down, let us assume that the cone at infinity is *regular* outside the origin: $\operatorname{sing} \mathbf{C}_\infty = \{ 0 \}$.

The two cones are related via the monotonicity of the area functional, which gives \begin{equation} \tag{1}\label{1} \lVert \mathbf{C}_0 \cap B_1 \rVert \leq \lVert \mathbf{C}_\infty \cap B_1 \rVert; \end{equation} equality occurs exactly when $\mathbf{C}_0 = M = \mathbf{C}_\infty$.

**Question.** Can anything else be said about the pair $(\mathbf{C}_0,\mathbf{C}_\infty)$? Is there a pair of cones $(\mathbf{C}_a,\mathbf{C}_b)$ which satisfies \eqref{1} strictly but is not a ‘blow-up/blow-down’ pair?

For clarification, some (families of) examples of pairs $(\mathbf{C}_0,\mathbf{C}_\infty)$ can be collated from the literature:

- Hardt and Simon constructed foliations which yield the pairs $(\Pi,\mathbf{C})$, where $\mathbf{C}$ is an arbitrary regular area-minimising cone and $\Pi$ is an $n$-dimensional plane;
- White constructed minimal surfaces $M$ with $0 \in \mathrm{sing} M$, and given blowdown cone $\mathbf{C}_\infty = \mathbf{C}$; this yields pairs $(\mathbf{C}_0,\mathbf{C}_\infty)$ where $\mathbf{C}_0$ is not a plane, but some hypotheses on $\mathbf{C}$ are required.

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