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In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1):

Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$ has finite index, there exists a compact set $C \subseteq M$ such that $M\setminus C$ is stable and there exists a positive function $u$ on $M$ such that $L u = 0$ on $M\setminus C$, where $L$ is the stability operator coming from the second variation of the area functional.

My question is if this statement is true in any dimension (assuming codimension $1$). I'm reading the proof and it seems to me that the argument is independent from the dimension, but maybe I'm wrong.

Any help will be very appreciated!

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  • $\begingroup$ I think this statement is probably false in higher dimesion since there is a singular stable minimal surface in $R^n$ with $n\geq8$: the Simons cone. $\endgroup$
    – Paul
    Commented Apr 24, 2019 at 20:02
  • $\begingroup$ @Paul what if I assume my mininal hypersurface to be smooth? $\endgroup$
    – Onil90
    Commented Apr 25, 2019 at 2:18
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    $\begingroup$ I am not sure I understand your comment, @Paul. It seems to me that the Simons cone does have the desired property outside a compact region, no? Having looked at the proof, it seems to me that at least the first property ought to hold in higher dimensions, namely that $M$ is stable outside a compact region. For the second, I am not sure I understand how to justify the use of the Harnack inequality, especially in situations where the area of $M$ might grow faster than polynomially. Have you thought about this point @Onil90? $\endgroup$
    – Leo Moos
    Commented Sep 26, 2020 at 16:51

1 Answer 1

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Yes, this statement is true in any dimension. The minimal surface is stable, which is equivalent to the stability operator L being non-negative. In fact, you can prove something similar for the general schrodinger operator. see Lemma3.10 on Stefano Pogola, Marco Rigoli, Alberto G.Setti's book: Vanishing and Finiteness Results in Geometric Analysis.

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