Viscosity solutions, ends and ideal boundaries

Abstract

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.