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I would like to know if there are any natural (e.g., physical) examples of non-flat, asymptotically flat manifold with non-positive sectional curvature? For example, any minimal surface of such a kind?

More generally, are there known examples of open (non-compact, connected) manifolds with non-positive sectional curvature that is of bounded geometry?

Thank you!

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For your first question: I do not know about "physical", but I know the following:

Theorem. Let $M$ be a finite volume noncompact hyperbolic manifold (i.e. a complete connected manifold of constant negative curvature). Then $M$ admits a metric of nonpositive curvature of bounded geometry which is complete and asymptotically flat.

See

B.Leeb, 3-manifolds with(out) metrics of nonpositive curvature, Inventiones Math., 1995.

Manifolds like this exist in all dimensions, for instance, every noncompact connected surface with finitely generated fundamental group and of negative Euler characteristic. (For instance, remove three points from the 2-sphere.)

As for your second question: Just take the Euclidean space. Or, if this is too simple, take one of the examples in part 1.

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  • $\begingroup$ Thank you so much!! $\endgroup$ Commented May 24, 2019 at 14:03
  • $\begingroup$ Hi Moishe, Thanks again for the nice reply! Sorry I'm not an expert on hyperbolic geometry or geometrisation of 3-manifolds. Could you show me a bit more on how to deduce your Theorem (i.e., Let $M$ be a finite volume noncompact hyperbolic manifold. Then $M$ admits a metric of nonpositive curvature of bounded geometry which is complete and asymptotically flat) from B. Leeb's paper? My guess is to use Theorem 3.2 and ''push the boundary of the compact [Haken] manifold to infinity''. But I wish to make sure things like this can be done rigorously. $\endgroup$ Commented May 24, 2019 at 18:50
  • $\begingroup$ @SiranVictorLi: See Proposition 2.3 in the paper: It constructs a metric on the compact manifold $M$ which is flat near the boundary. Then attach to the result the product manifold $\partial M\times [0,\infty)$ (with the product metric which is necessarily flat). $\endgroup$ Commented May 24, 2019 at 21:19

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