Steiner minimal trees in small neighbourhoods of points in Riemannian manifolds

In contrast to the Euclidean case, almost no Steiner minimal trees with concrete boundaries on Riemannian manifolds are known. A result describing the types of Steiner minimal trees on a Riemannian manifold for arbitrary small boundaries is obtained. As a consequence, it is shown that for sufficiently small regular $n$-gons with $n\geqslant 7$ their boundaries without a longest side are Steiner minimal trees.
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