Negatively curved manifolds, elliptic operators, and the Martin boundary.

*(English)*Zbl 0652.31008Let M denote a complete, simply connected Riemannian manifold whose sectional curvature is bounded between two negative constants. M admits a sphere at infinity \(S_{\infty}M\) whose points consist of asymptotic unit speed geodesics of M; two unit speed geodesics \(\gamma\), \(\sigma\) of M are asymptotic if d(\(\gamma\) t,\(\sigma\) t)\(\leq c\) for all \(t\geq 0\) and some constant \(c>0\). Alternatively, one can define \(S_{\infty}M\) as follows: Fix a point \(x_ 0\in M\). A sequence \(\sigma =\{x_ n\}\) in M is admissible if \(d(x_ n,x_ 0)\to \infty\) and if the functions \(f_ n(x)=d(x,x_ n)-d(x_ 0,x_ n)\) converge to a function \(f_{\sigma}(x): M\to {\mathbb{R}}\) as \(n\to \infty\). Two admissible sequences \(\sigma\), \(\sigma\) ’ are equivalent if the corresponding functions \(f_{\sigma}\), \(f_{\sigma '}\) are equal. One can show that a sequence \(\sigma =\{x_ n\}\) is admissible if and only if \(\sigma\) converges to a point z in \(S_{\infty}M\) in the usual sense (the angle subtended at \(x_ 0\) by \(x_ n\) and z converges to zero). In this case the function \(f_{\sigma}\) is called the Busemann function determined by z and \(x_ 0.\)

Now let \({\mathcal L}\) be a second order elliptic operator on functions from M to \({\mathbb{R}}\) such that for some positive number \(\epsilon\) there exists a positive function f: \(M\to {\mathbb{R}}\) that is superharmonic with respect to \({\mathcal L}+\epsilon I\). Let G: \(M\times M\to {\mathbb{R}}\) be the corresponding \({\mathcal L}\)-Green’s function, and let \(G_ x: y\to G(y,x)\) denote the \({\mathcal L}\)-harmonic function with singularity at x. Fix a point \(x_ 0\) in M. A sequence \(\sigma =\{x_ n\}\) in M is \({\mathcal L}\)- admissible if \(d(x_ n,x_ 0)\to \infty\) and if the functions \(K_ n(x)=G_{x_ n}(x)/G_{x_ n}(x_ 0)\) converge to a function \(K_{\sigma}(x)\) as \(n\to \infty\). \({\mathcal L}\)-admissible sequences \(\sigma\), \(\sigma\) ’ are equivalent if the corresponding functions \(K_{\sigma}\), \(K_{\sigma '}\) are equal. The points of the \({\mathcal L}\)- Martin boundary are defined to be the equivalence classes of \({\mathcal L}\)- admissible sequences \(\sigma\).

Theorem: Let M and \({\mathcal L}\) be as above. Let \(\sigma =\{x_ n\}\) be a sequence in M converging to a point z in \(S_{\infty}M\). Then \(\sigma\) is an \({\mathcal L}\)-admissible sequence, and the corresponding map \(z\to K_{\sigma}\) is a homeomorphism of \(S_{\infty}M\) onto the \({\mathcal L}\)- Martin boundary. Corollary (Dirichlet problem): Let \({\mathcal L}\) and M be as above, and suppose further that \({\mathcal L}(1)=0\) and \(G_ x=0(1)\) at infinity in M for all x in M. Then for every continuous real valued function f: \(S_{\infty}M\to {\mathbb{R}}\) there exists a unique function u: \(M\cup S_{\infty}M\to {\mathbb{R}}\) with \(u=f\) on \(S_{\infty}M\) and \({\mathcal L}(u)=0\) on M.

Example: Let \({\mathcal L}\) be the Laplacian acting on functions f: \(M\to {\mathbb{R}}\), where M is as above with sectional curvature \(K\leq -a\) \(2<0\). Let \(\delta =a(n-1)/2\) and let \(f(x)=e^{-\delta p(x)}\), where \(p(x)=d(x,x_ 0)\). Then \(\Delta f+\alpha f\leq 0\) if \(0<\alpha <\delta\) 2. In this special case the two results above are due to M. T. Anderson and R. Schoen [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)].

The results in this paper follow from potential theoretic methods and are actually true in greater generality than stated above. In particular one can replace the bounded negative curvature condition by geometric conditions that are invariant under quasi-isometries. The author also applies his results and methods to elliptic operators on domains on \({\mathbb{R}}^ n \)and to random walks on discrete sets.

Now let \({\mathcal L}\) be a second order elliptic operator on functions from M to \({\mathbb{R}}\) such that for some positive number \(\epsilon\) there exists a positive function f: \(M\to {\mathbb{R}}\) that is superharmonic with respect to \({\mathcal L}+\epsilon I\). Let G: \(M\times M\to {\mathbb{R}}\) be the corresponding \({\mathcal L}\)-Green’s function, and let \(G_ x: y\to G(y,x)\) denote the \({\mathcal L}\)-harmonic function with singularity at x. Fix a point \(x_ 0\) in M. A sequence \(\sigma =\{x_ n\}\) in M is \({\mathcal L}\)- admissible if \(d(x_ n,x_ 0)\to \infty\) and if the functions \(K_ n(x)=G_{x_ n}(x)/G_{x_ n}(x_ 0)\) converge to a function \(K_{\sigma}(x)\) as \(n\to \infty\). \({\mathcal L}\)-admissible sequences \(\sigma\), \(\sigma\) ’ are equivalent if the corresponding functions \(K_{\sigma}\), \(K_{\sigma '}\) are equal. The points of the \({\mathcal L}\)- Martin boundary are defined to be the equivalence classes of \({\mathcal L}\)- admissible sequences \(\sigma\).

Theorem: Let M and \({\mathcal L}\) be as above. Let \(\sigma =\{x_ n\}\) be a sequence in M converging to a point z in \(S_{\infty}M\). Then \(\sigma\) is an \({\mathcal L}\)-admissible sequence, and the corresponding map \(z\to K_{\sigma}\) is a homeomorphism of \(S_{\infty}M\) onto the \({\mathcal L}\)- Martin boundary. Corollary (Dirichlet problem): Let \({\mathcal L}\) and M be as above, and suppose further that \({\mathcal L}(1)=0\) and \(G_ x=0(1)\) at infinity in M for all x in M. Then for every continuous real valued function f: \(S_{\infty}M\to {\mathbb{R}}\) there exists a unique function u: \(M\cup S_{\infty}M\to {\mathbb{R}}\) with \(u=f\) on \(S_{\infty}M\) and \({\mathcal L}(u)=0\) on M.

Example: Let \({\mathcal L}\) be the Laplacian acting on functions f: \(M\to {\mathbb{R}}\), where M is as above with sectional curvature \(K\leq -a\) \(2<0\). Let \(\delta =a(n-1)/2\) and let \(f(x)=e^{-\delta p(x)}\), where \(p(x)=d(x,x_ 0)\). Then \(\Delta f+\alpha f\leq 0\) if \(0<\alpha <\delta\) 2. In this special case the two results above are due to M. T. Anderson and R. Schoen [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)].

The results in this paper follow from potential theoretic methods and are actually true in greater generality than stated above. In particular one can replace the bounded negative curvature condition by geometric conditions that are invariant under quasi-isometries. The author also applies his results and methods to elliptic operators on domains on \({\mathbb{R}}^ n \)and to random walks on discrete sets.

Reviewer: P.Eberlein

##### MSC:

31C35 | Martin boundary theory |

31C12 | Potential theory on Riemannian manifolds and other spaces |

53C99 | Global differential geometry |