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Viscosity solutions, ends and ideal boundary

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.