Limits of Dirichlet finite functions along curves (Q793192)

Let R be a noncompact connected real analytic Riemannian n-manifold (perhaps without boundary). Let \(R^*\) be the Royden compactification of R (to which Dirichlet finite functions extend continuously) and let \(\Delta =R^*-R\) be the harmonic part of R. Let \(\tilde M\)(R) be the space of Tonnelli functions on R with finite Dirichlet integrals \(D_ R(f)=\int_{R}df\quad \wedge \quad *\quad df\quad<\quad \infty.\) If H is a family of locally rectifiable curves in R, a non-negative Borel measurable function \(\rho\) is admissible with respect to H if \(\int_{\gamma}\rho ds\geq 1\) for every \(\gamma\in H\) and the modulus of H is given by \(mod H=\inf \{\int_{R}\rho^ 2\quad dV:\quad \rho \quad admissible\}.\) A property is said to hold H everywhere if it holds except on a set \(H_ 0\) of modulus zero. Let B be a fixed ball in R and let G be the family of all curves in R-B joining the boundary of B to the ideal boundary of R. If \(f\in \tilde M(R)\), then \(\lim_{t\to \infty}f(\gamma(t))\) exists for almost all \(\gamma\in G\). The author's main result is then that for \(f\in \tilde M(R)\), \(f(\gamma)=0\) for almost all \(\gamma\in G\) if and only if \(f|_{\Delta}=0\).