Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N . For $$1\leqslant k\leqslant n$$ 1 ⩽ k ⩽ n , we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup $${\mathbb {V}}$$ V of $${\mathbb {H}}^n$$ H n can be extended to an intrinsic $$L'$$ L ′ -Lipschitz graph over the entire subgroup $${\mathbb {V}}$$ V , where $$L'$$ L ′ depends only on L, k, and n. We further prove that 1-dimensional intrinsic 1-Lipschitz graphs in $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N , admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $${\mathbb {H}}^1$$ H 1 . The main difference to this case arises from the fact that for $$1\leqslant k<n$$ 1 ⩽ k < n , the complementary vertical subgroups of k-dimensional horizontal subgroups in $${\mathbb {H}}^n$$ H n are not commutative.

A geometric definition of the Mañé-Mather set and a Theorem of Marie-Claude Arnaud.

We study some properties of Lipschitz exact Lagrangian manifolds isotopic to the zero section. We prove that if such a manifold is invariant under an optical Hamiltonian, then it must be a Lipschitz graph. This extends a recent result of Marie–Claude Arnaud. We also obtain a new geometric description of the Mañé–Mather invariant set.