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Let $\Sigma \subset \mathbb{R}^{n+1}$ be a complete and connected minimal submanifold (with or without boundary), i.e. its mean curvature $H$ vanishes everywhere. Assume also that there exists an open subset $A \subset \Sigma$ contained in a hyperplane, i.e. where the second fundamental form vanishes.

Can we say that the second fundamental form vanishes everywhere in $\Sigma$ ?

Any help would be very appreciated.

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    Do you know that minimal submanifolds are real analytic? – 2017-01-13
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    @MoisheCohen I can believe it, since the strong relation between minimal surfaces and harmonic functions, but I didn't know that. – 2017-01-13

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Lemma 1 here: Minimal submanifolds are real analytic.