# Characterizing Euclidean 3-space

In my previous post about the recent preprint of Jiang Wang we saw the following characterization of Euclidean 3-space among all contractible 3-manifolds: It is the only one that admits a complete Riemannian metric of non-negative scalar curvature.

Today a preprint was posted on the arXiv:2105.09035 by Bargagnati and Frigerio providing another characterization: The only contractible 3-manifold with vanishing simplicial volume is Euclidean 3-space (and any other contractible 3-manifold has infinite simplicial volume).

Let me quickly recall the definition of simplicial volume. Let M be an orientable n-manifold and denote by [M] its fundamental class in its locally finite homology. Then $\|M\| := \inf \big\{\sum |a_\sigma|\colon x = \sum a_\sigma \sigma \in C_n^{\mathrm{lf}}(M;\mathbb{R})\ \& \ [x] = [M]\big\}\,.$

As a corollary they also obtain the following: Euclidean 3-space is the only contractible 3-manifold that supports a complete Riemannian metric $$g$$ of finite volume and with Ricci curvature uniformly bounded from below, i.e. $$\mathrm{Ric}_g \ge -kg$$ with $$k > 0$$.

Finally, note that Bargagnati and Frigerio further prove that their main result is special to dimension 3, i.e., in every higher dimension there exists a contractible, smooth manifold not homeomorphic to Euclidean space but with vanishing simplicial volume.

What about the other characterizations of Euclidean space in higher dimensions?

• Is Euclidean space the only contractible manifold supporting a complete Riemannian metric of finite volume and with Ricci curvature uniformly bounded from below?
• Is Euclidean space the only contractible manifold with vanishing minimal volume? (I have not mentioned this above, but this is another corollary that Bargagnati and Frigerio get in dimension 3.)

Regarding the characterization proven by Jiang Wang: After talking to Bernhard Hanke and to Rudolf Zeidler about this, I think that this should actually fail in higher dimensions, i.e., there should be higher-dimensional, contractible manifolds admitting a complete Riemannian metric of (uniform) positive scalar curvature.

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