Can the complete simply-connected surface with constant Gauss curvature -1 be embedded smoothly in the 5-dimensional Euclidean space?
Can the complete simply-connected surface with constant Gauss curvature -1 be immersed smoothly in the 4-dimensional Euclidean space?
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1$\begingroup$ I suppose you already know that one can embed hyperbolic space in $\mathbb{R}^3$ with the Minkowski metric and that the Nash embedding theorem gives an embedding for any surface into Euclidean space. The dimension bound on wikipedia gives just $n \le 51$ so that is pretty far off the $4$ or $5$ dimensions you are hoping for. $\endgroup$– quaragueCommented Jul 8, 2019 at 8:51
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6$\begingroup$ See math.stackexchange.com/questions/1528046/… and references therein. $\endgroup$– Moishe KohanCommented Jul 8, 2019 at 9:28
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2$\begingroup$ A piecewise analytic map (but apparently not smooth) is given here: mathscinet.ams.org/mathscinet-getitem?mr=1025303 $\endgroup$– Ian AgolCommented Jul 8, 2019 at 15:43
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$\begingroup$ I care more about the smooth embedding and immersion, so I edit the question. I know the Nash embedding theorem, but I never heard how to embed hyperbolic space in R^3 with the Minkowski metric. Could you please explain it more precisely? What's more, are there any relationship between the C^1 embedding in R^3 and the piecewise analytic immersion in R^4? Thanks for every helpful answers. $\endgroup$– 011000Commented Jul 9, 2019 at 4:25
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2$\begingroup$ As of the very comprehensive 2001 survey of Borisenko both questions seem to be open. For Q1 if you replace "embed" by "immerse" then the result holds due to Rozendorn. (The original paper is in Russian (see Borisenko for reference) but the construction is outlined in Rozendorn's 1992 survey.) And Ian Agol already mentioned the result of Sabitov for Q2 which is only $C^{0,1}$ globally but piecewise analytic. $\endgroup$– Willie WongCommented Jul 9, 2019 at 15:11
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