On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold $(M,g)$, there are three kinds of objects that have been studied extensively:
$\bullet $ Viscosity solutions to the Hamilton–Jacobi equation determined by the Riemannian metric;
$\bullet $ Ends introduced by Freudenthal and more general other remainders from compactification theory;
$\bullet $ Various kinds of ideal boundaries introduced by Gromov.
In this paper, we will present some initial relationship among these three kinds of objects and some related topics are also considered.
"Viscosity solutions, ends and ideal boundaries." Illinois J. Math. 60 (2) 459 - 480, Summer 2016. https://doi.org/10.1215/ijm/1499760017