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In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal submanifolds can be reduced to the study of minimal cones and hence to compact minimal subsminifolds of the sphere.

I understand this might be one of those results relating the so called tangent cones at infinity to cones over compact submanifolds of spheres, but I would like some more clarification.

  1. What is the definition of the tangent cone at infinity of a minimal submanifold of dimension $k$ in $\mathbb{R}^{m+k}$? What is precisely the blow down?

  2. What is the dimension of this tangent cone at infinity? Under which conditions is this cone smooth outside the tip?

  3. What is the definition of the minimal cones (not tangent at infinity) and how do they relate to compact submanifolds of the sphere?

  4. I understand the author uses an identification of the tangent cone at infinity of the minimal submanifold of $\mathbb{R}^{m+k}$ with the cone over the compact submanifold of $S^{m+k-1}$. But what is the dimension of this compact submanifold so that its cone will "coincide" with the cone at infinity in $\mathbb{R}^{m+k}$?

  5. Are there any specific references for that? Such things appear here and there, but I have never seen a proof neither a narrow reference. I have looked and got lost in Federer's book.

I would be very happy if I can understand precisely all these identifications and definitions.

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    $\begingroup$ Presumably Fischer-Colbrie means tangent cones at a singularity not tangent cones at infinity here. It would help if you could clean your question up to ask one specific question. It's a standard fact in GMT that if you blow up at a singular point of a $k$-dimensional minimal surface in $R^{n+k}$ you converge to a stationary integral $k$-dimensional varifold in $R^{n+k}$ that's invariant under dilation. In general, this varifold will be very singular. But you can regard it as a varifold cone over a $k-1$-dimensional stationary integral varifold in $S^{n+k-1}$. $\endgroup$
    – Otis Chodosh
    Commented Jul 24, 2023 at 17:06
  • $\begingroup$ So if your question is just, what's the dimension of the tangent cone at a singularity, it's the same as the dimension of the minimal surface. $\endgroup$
    – Otis Chodosh
    Commented Jul 24, 2023 at 17:07
  • $\begingroup$ @OtisChodosh: Thanks for the reply. Maybe my question is not really about the dimension, I think I wanted just to confirm it is the same dim. It is really about the definition of the tangent cone at a singularity as you say, also of the tangent cone at infinity, since I keep encountering this expression here and there but never the precise definition and more importantly: where can I read the proofs of these "standard facts of GMT"? $\endgroup$
    – Cris.giansu
    Commented Jul 24, 2023 at 20:24
  • $\begingroup$ Another thing, by a blow up do you simply mean multiplying the metric of the manifold by some r_i that goes to infinity as i grows? $\endgroup$
    – Cris.giansu
    Commented Jul 24, 2023 at 20:26
  • $\begingroup$ A good reference is Leon Simons GMT, in the typewriter (1st edition) version you can see Theorem 19.3. $\endgroup$
    – Otis Chodosh
    Commented Jul 24, 2023 at 21:02

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