Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere
For a Riemannian three-manifold $(\Sigma,g)$ with negative Einstein endomorphism $\mathbb{M}=-G$, the Gauss equation of a hypersurface of $\mathbb{R}^4$ reads $\mathbb{M}=\mathrm{adj}\,K$, and on $\{\det\mathbb{M}>0\}$ it is solved identically by $\tilde K=\sqrt{\det\mathbb{M}}\,\mathbb{M}^{-1}$. Realizability with nondegenerate second fundamental form is therefore the single equation $F(g)=\mathrm{Cod}(\tilde K(g))=0$. The reduction is classical (Beez, Killing, Thomas) and the zero set has been characterized intrinsically (Rivertz; Agaoka and Hashinaga); we study $F$ as a nonlinear operator…
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