Given a Lie algebroid structure on a vector bundle $A$ and a leaf $L$ of this Lie algebroid, a natural question is whether all nearby Lie algebroid structures on A have a leaf which is diffeomorphic to $L$. This question was answered by M. Crainic and R. Fernandes. In this talk, I will show that this question is an instance of a general question about differential graded Lie algebras: Given a differential graded Lie algebra $\mathfrak{g}$, a differential graded Lie subalgebra $\mathfrak{h}$, and a Maurer-Cartan element $Q$ of the subalgebra, are all Maurer-Cartan elements of $\mathfrak{g}$ near $Q$ gauge equivalent to an element of $\mathfrak{h}$? In the case that $\mathfrak{h}$ has finite codimension in $\mathfrak{g}$, I will give a sufficient criterium for a positive answer to the question. As a consequence, I will mention some stability results for zero-dimensional leaves of various geometric structures that can be obtained from this.

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