Constant extrinsic curvature surfaces in space forms are best understood as Sasakian objects living in the unitary bundle. In this framework they become bilegendrian surfaces. We discuss the insights this perspective provides on the study of constant extrinsic curvature surfaces.

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In this talk we study applications of the Kulkarni-Pinkall form to the study of locally strictly convex immersions in \(\mathbb{H}^3\). We deduce a new a priori estimate which in turn allows us to completely solve the asymptotic Plateau problem for \(k\)-surfaces in hyperbolic space as formulated by Labourie. This work has interesting intersections with a paper of Espinar-Gálvez-Mira. This work appears in https://arxiv.org/abs/2104.03181.

In arXiv:1010.1879 we construct an integer valued degree theory for compact, immersed hypersurfaces of constant curvature in general Riemannian manifolds, and we show how this degree theory may be used to prove existence results such hypersurfaces. Joint work with Harold Rosenberg.

We show how barrier techniques are used to solve the Plateau problem for general curvature functions in general manifolds.
arXiv:1008.3545

We prove the existence of constant curvature hypersurfaces in Hadamard manifolds subject satisfying topological/boundary conditions.

We discuss how the theory of pseudo-holomorphic curves may be used to study constant Gaussian curvature surfaces in three dimensional manifolds.