Viscosity solutions, ends and ideal boundaries (Q2362694)

By analogy with Aubry-Mather theory, the author extends his work [Proc. Am. Math. Soc. 143, No. 10, 4423--4431 (2015; Zbl 1326.53052)] to provide geometric interpretations of viscosity solutions to the eikonal equation. For \((M,g)\) a complete, non-compact Riemannian manifold without boundary, the author showed in unpublished work with J. Cheng that functions of the form \[ dl(x) = \underset{n \to \infty}{\lim} \; d(x, K_n) - d(y, K_n)\eqno{(*)} \] for \(y \in M\) fixed and \(K_n\) compact are solutions of \[ \left| \bigtriangledown_g u \right|_g = 1\eqno{(**)} \] on \(M\). The paper focuses on the opposite question showing that up to a constant all solutions of \((\ast \ast)\) are of the form \((\ast)\). Functions of the form \((\ast)\) are generalizations of Busemann functions \[ b_{\gamma}(x) = \underset{t \to \infty}{\lim} \; d\left(x, \gamma(t) \right) - t \] for \(\gamma\) a ray in \(M\). Such functions provide a bridge to two notions of remainder in the context of compactification: the metric notion of ideal boundary due to Gromov and the topological notion of end due to Freudenthal. The author provides two proofs of the correspondence between \((\ast)\) and \((\ast \ast)\). One uses the Hopf-Lax theorem; the other exploits the correspondence between viscosity solutions and variational solutions of Chaperon-Sikorav-Viterbo [\textit{T. Zhukovskaya}, J. Math. Sci., New York 82, No. 5, 3737--3746 (1996; Zbl 0901.58019)].