Georgia Institute of Technology College of Sciences

Isometric embeddings between a domain manifold and a target manifold are differentiable maps f such that the pullback of the target metric h coincides with the metric g in the domain manifold. This problem can also be formulated as a non-linear PDE via $\nabla f^{\top} h \nabla f = g$. In the case of contact manifolds, it is additionally required that the embedding preserves a certain restriction on the tangent bundle.

We prove that the Nash iteration scheme can be quantified in order to construct infinitely many $C^{1,\alpha}$-isometric embeddings for contact manifolds. In this way, we extend an existing result regarding non-uniqueness for $C^1$ regularity. The strategy of the proof follows a paper by Conti, De Lellis and Szekelyhidi Jr. on the Riemannian case, which is built on the Nash-Kuiper scheme. The main difficulty in our case is to keep the additional linear constraint coming from the contact setting along the iteration procedure.

In the larger program of a quantitative analysis of isometric embeddings between sub-Riemannian manifolds, our result can be seen as an important first step. Another aspect is the flexibility of this convex integration method: the geometric constraint coming from the contact condition is just one special case of a (potentially large) class of admissible constraints, under which this scheme can still be applied.