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Consider two $n$-dimensional minimal surfaces without boundary. Suppose they are embedded in some $\mathbb{R}^m$ in such a way that they intersect transversally. Is their intersection a minimal surface? For example, when two hyperspheres overlap, their intersection is another, lower dimensional hypersphere, which is minimal (notice I am only counting the intersection of the surfaces of the spheres, not the interiors). Is this true in general?

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    $\begingroup$ Is the hypersphere a minimal surface? $\endgroup$ – Fan Zheng Nov 3 '17 at 5:07
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No: consider the catenoid in euclidean 3-space and intersect it with a plane which is perpendicular to the symmetry axis of the catenoid. The intersection is a circle, which is not s geodesic (minimal surface of dimension 1).

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