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We give a sufficient condition for branched minimal immersions of spheresinto ellipsoids to be embedded: we show that if the coordinate functions of thebranched minimal immersion are first or second eigenfunctions with respect to anatural metric on the sphere involved in this problem, called critical metric,then it is an embedding into a 2-dimensional ellipsoid or an immersion into a3-dimensional ellipsoid. If in addition this critical metric is rotationallysymmetric with respect to an axis and symmetric with respect to the orthogonalplane then the immersion is an embedding. We also give a construction ofnon-planar minimal spheres into 3-dimensional ellipsoids such that thecoordinate functions are first and second eigenfunctions with respect to thecritical metric obtained by maximization of linear combinations of first andsecond (area renormalized) Laplace eigenvalues among metrics on the spherehaving the aforementionned symmetry constraints or not.

Laplace eigenvalues and non-planar minimal spheres into 3-dimensional ellipsoids