Quaternions, Monge-Amp\`ere structures and $k$-surfaces
Graham Smith
TL;DR
This paper presents a quaternionic reformulation of the theory of immersed surfaces with prescribed extrinsic curvature, simplifying proofs and paving the way for higher-dimensional generalizations in hyperbolic geometry and related fields.
Contribution
The paper introduces a quaternionic approach to surface theory with prescribed curvature, providing simpler proofs and extending the framework to higher dimensions.
Findings
Simplified proofs of existing results
Quaternionic reformulation of surface theory
Framework for higher-dimensional generalizations
Abstract
In [15] Labourie develops a theory of immersed surfaces of prescribed extrinsic curvature which has since found widespread applications in hyperbolic geometry, general relativity, Teichm\"uller theory, and so on. In this chapter, we present a quaternionic reformulation of these ideas. This yields simpler proofs of the main results whilst pointing towards the higher-dimensional generalisation studied by the author in [25].
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Algebraic Geometry and Number Theory