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We prove an equivariant implicit function theorem for variational problemsthat are invariant under a varying symmetry group (corresponding to a bundle ofLie groups). Motivated by applications to families of geometric variationalproblems lacking regularity, several non-smooth extensions of the result arediscussed. Among such applications is the submanifold problem of deforming theambient metric preserving a given variational property of a prescribed familyof submanifolds, e.g., constant mean curvature, up to the action of thecorresponding ambient isometry groups.

Deforming solutions of geometric variational problems with varying symmetry groups