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Consider $(\tilde{M},g)$ a riemannian manifold and $M \subset \tilde{M}$ riemannian submanifold.

Is it true that if $M$ is a minimal submanifold of $\tilde{M}$ then for every $p \in M$ there exists a neighborhood $W$ of $p$ in $\tilde{M}$ such that $V=W\cap M$ has least area among every $\Omega \subset W$ with $\partial \Omega = \partial V$?

I've been thinking about it, I think it is true but I don't know how to prove.

If it's true, how should I go about proving it?

Arctic Char
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The question was re-asked in Mathoverflow and get a good answer. The answer is yes and the proof uses a calibration argument. Reader who are interested only in $\mathbb R^3$ may see the following less technical answer.

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    In the case of $\Bbb R^3$, the calibration argument is given explicitly as an exercise in my Multivariable Mathematics book. See Exercise 22, p. 392. It's a direct application of Stokes's Theorem, just as Robert indicates in his general answer. – Ted Shifrin Jul 21 '20 at 22:07
  • Need to assume orientability of $\Omega$ for this argument to apply. – Luis A. Florit Sep 13 '24 at 17:49