# Tangent cones at zero and infinity to minimal surfaces

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 $$$$\tag{1}\label{1} \lVert \mathbf{C}_0 \cap B_1 \rVert \leq \lVert \mathbf{C}_\infty \cap B_1 \rVert;$$$$ equality occurs exactly when $$\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.
• An obvious example is two multiplicity one hyperplanes that are not parallel. Actually, this works for any pair of cones with the same density which are not equal. Commented Sep 5, 2022 at 17:22
• Yeah, you're right - I should have commented on equality in the monotonicity inequality. I'll fix it in a second. Commented Sep 5, 2022 at 17:29
• I think it's a an open problem to find M with an isolated singularity but no boundary (other than a cone). One expects this to occur as blowups of intermediate scales in degenerating min surf in R8 for example. See the work of Edelen. Commented Sep 6, 2022 at 2:57
• @OtisChodosh I'll take a look at Nick's stuff, thanks for the pointer! Would you expect the blow-up at the singularity to be related with $\mathbf{C}_\infty$? Say some constraints via topology, or the Morse index of their links? Commented Sep 6, 2022 at 3:06
• I guess the simplest example is two different Lawson's cones $(C^{p, q}, C^{p', q'})$, here $p'<p\leq q<q'$, $p+q = p'+q'\geq 8$. Note that two Lawson's cones have the same density at origin iff they are equivalent by an ambient isometry. On the other hand, [Simon-Solomon '86] proved that any stationary integral varifold asymptotic to a multiplicity one Lawson's cone $C$ near infinity is translation of either $C$ itself or the Hardt-Simon foliation on one-side of $C$, thus, the only blow-up-blow-down pair with $C_\infty$ to be a Lawson's cone must have $C_0$ planar. Commented Aug 14, 2023 at 5:50

All we have to do is find any singular minimal cone that is not the limit of smooth embedded surfaces. This is because we will take $$\mathbf{C}_a =$$ a plane and a plane has density 1 and a singular cone must have density $$> 1$$.
• Thanks for the answer, but it's not quite what I was looking for. After your comments, I narrowed the question to consider only cones $\mathbf{C}_\infty$ with an isolated singularity, essentially to avoid examples such as the one you gave. As for the first question being open-ended, I disagree: it's a yes/no question. Now you could criticize that it's imprecise, but that's deliberate. What the question boils down to is this. If I have a blowdown cone $\mathbf{C}_\infty$, does this constrain what tangent cones I can get at singularities? What if I have the tangent cone $\mathbf{C}_0$ at [...] Commented Sep 6, 2022 at 17:45
• a singularity, does this constrain the blowdown cones you can get? (I mean, beyond the constraints for their densities that the monotonicity formula gives.) The first question is deliberately imprecise because I am interested in any answers: is there a topological or a variational relationship between the pair $(\mathbf{C}_0,\mathbf{C}_\infty)$? Can $\mathbf{C}_0$ have a 'bad' singular set if $\mathrm{sing} \, \mathbf{C}_\infty = \{0 \}$? Whether you work in the smoothly embedded class, or with area-minimizing currents or stationary varifolds, I'm interested in any information at all. Commented Sep 6, 2022 at 17:49