I wonder if someone can clarify whether it is known that every closed, orientable surface (2-manifold) has a smooth isometric immersion in $\mathbb{R}^4$?
This topic has been discussed rather thoroughly in the earlier MO question, "Nash embedding theorem for 2D manifolds." However, I am puzzled by two claims that seem at odds, but are surely not—rather I just don't understand the terminology.
First, Gromov established that every compact surface has a smooth embedding in $\mathbb{R}^5$. This is from his 1986 book, Partial Differential Relations. See the above MO question for a quote.
Second, as user jc said, "Poznyak proved that $\mathbb{R}^4$ works for any compact part of a complete surface." Here is a more precise statement of Poznyak's result, from the book by Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces (p.40):
Theorem 2.3.1. Let $g$ be a smooth complete metric defined in $\mathbb{R}^2$. Then for any compact set $\Omega$ in $\mathbb{R}^2$, $(\Omega,g)$ admits a smooth isometric immersion in $\mathbb{R}^4$.
Since Poznyak's $\mathbb{R}^4$ result is from 1973, and Gromov's $\mathbb{R}^5$ from 1986, it must be that Poznyak's theorem does not improve Gromov's to $\mathbb{R}^4$. Is this because "a compact part of a complete surface" is not the same as a "compact surface"? Can anyone explain this to me? Thanks!
E.G. Poznyak, "Isometric immersions of two-dimensional Riemannian metrics in Euclidean space." Russian Math. Surveys, 28:4 (1973), pp. 47–77. Uspekhi Mat. Nauk, 28:4 (1973), pp. 47–76.
Answered. Robert Bryant resolved my confusion: "A compact surface without boundary can't be identified with a subset of the plane." It is my understanding now that it remains an open question of whether or not every compact surface (without boundary) has a smooth immersion in $\mathbb{R}^4$: neither Gromov's nor Poznyak's results settle this question, and as far as I know there is no known counterexample.
I'm adding the open-problem
tag, because although Robert completely answered my specific
question, the question suggested by the title is open.