Smooth isometric immersion of every closed Riemannian surface into R^4

Theorem 1.1: every closed smooth Riemannian surface (compact, without boundary, not necessarily orientable or connected, any Gaussian curvature) admits a $C^\infty$ isometric immersion into $\mathbb R^4$; self-intersections are allowed. This lowers Gromov's $\mathbb R^5$ immersion dimension for closed surfaces to four. Not shown: embeddings, noncompact or bounded surfaces (the paper says it treats the compact boundaryless case of Gromov's question), or analytic regularity.

Theorem 1.1: every closed smooth Riemannian surface (compact, without boundary, not necessarily orientable or connected, any Gaussian curvature) admits a $C^\infty$ isometric immersion into $\mathbb R^4$; self-intersections are allowed. This lowers Gromov's $\mathbb R^5$ immersion dimension for closed surfaces to four. Not shown: embeddings, noncompact or bounded surfaces (the paper says it treats the compact boundaryless case of Gromov's question), or analytic regularity.