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Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds. (English) Zbl 0541.53035

The author proves the existence of nontrivial bounded harmonic functions on certain classes of noncompact complete Riemannian manifolds by posing and solving under suitable conditions an asymptotic Dirichlet problem. The asymptotic Dirichlet problem is considered in two situations: a) manifolds of arbitrary dimension with a ”pole”; b) finitely connected surfaces whose ends are either ”standard expanding” or ”standard shrinking”. In each case one attempts to construct harmonic functions with given (asymptotic) boundary values by using the classical Perron method and proving the existence of appropriate barrier functions.
A Riemannian manifold M is said to have a pole at a point p if the exponential map at p is a diffeomorphism of \(T_ pM\) onto M. In particular such a manifold M is diffeomorphic to Euclidean space. Examples include but are far from limited to a) any complete, simply connected manifold of nonpositive sectional curvature (where any point is a pole) such as the hyperbolic plane regarded as the interior of the unit disk in the plane with a Riemannian metric of curvature -1 or b) certain manifolds of positive curvature such as the rotational paraboloid \(z=x^ 2+y^ 2\), where (0,0,0) is a pole. For manifolds with a pole p there is by definition for each point \(q\neq p\) in M a unique unit speed geodesic \(\gamma_{pq}\) joining p to q whose initial velocity will be denoted by V(p,q). Let \(S_ p\) denote the tangent vectors to M at p of length 1. We say that a moving point q converges to \(\gamma_ v(\infty)\), written \(q\to \gamma_ v(\infty)\), if d(p,q)\(\to \infty\) and the angle subtended by v and V(p,q)\(\to 0\). The asymptotic Dirichlet problem is to find for each continuous function \(\phi:S_ p\to {\mathbb{R}}\) a harmonic function u:\(M\to R\) such that u(q)\(\to \phi(v)\) as \(q\to \gamma_ v(\infty)\) for each \(v\in S_ p\). This problem may be rephrased for simply connected manifolds of nonpositive sectional curvature, which admit an infinite boundary \(M(\infty)\) consisting of asymptotic geodesics: given a continuous function \(\phi\) :\(M(\infty)\to {\mathbb{R}}\) find a harmonic function u:\(M\to R\) such that u(q)\(\to \phi(x)\) as \(q\to x\) (in a suitable topology) for each point \(x\in M(\infty)\). If M is the hyperbolic plane viewed as the interior of the unit disk, then \(M(\infty)\) is the unit circle \(S^ 1\) and the asymptotic Dirichlet problem becomes the classical Dirichlet problem for the disk. Unfortunately a general Riemannian manifold with a pole does not come equipped with a Poisson formula for generating harmonic functions with prescribed boundary values, except in special cases such as the symmetric spaces of noncompact type. In fact if M is n-dimensional Euclidean space with the standard flat metric then Liouville’s theorem says that the asymptotic Dirichlet problem has no solution.
Now let M be a manifold with a pole p, and let \(\phi:S_ p\to {\mathbb{R}}\) be a given continuous function. Let \(F=\{f:M\to R:f\) is subharmonic and lim sup f(q)\(\leq \phi(v)\) as \(q\to \gamma_ v(\infty)\) for all \(v\in S_ p\}\). Define \(u(q)=\sup \{f(q):\quad f\in F\}.\) It is well known that u is harmonic and the problem is to show that u satisfies the asymptotic boundary conditions. A function B:\(M\to R\) is called a barrier at v with angle \(\delta\) if 1) B is subharmonic; 2) \(B\leq 0\) and B(q)\(\to 0\) as \(q\to \gamma_ v(\infty)\); 3) There exists \(\eta>0\) such that lim sup B(q)\(\leq -\eta\) as \(q\to \gamma_ w(\infty)\) for any \(w\in S_ p\) with \(\sphericalangle_ p(v,w)>\delta\). Theorem. Suppose for any \(v\in S_ p\) and any sufficiently small \(\delta>0\) there is a barrier B at v with angle \(\delta\). Then lim u(q)\(=\phi(v)\) as \(q\to \gamma_ v(\infty)\) for any \(v\in S_ p\). In particular M possesses nonconstant bounded harmonic functions.
The author goes on to show that this barrier existence condition is satisfied in each of the following cases for a manifold with a pole p. a) Let M be rotationally symmetric and suppose that outside a compact set the radial curvature \(\leq -A/r^ 2\log r\) for some constant \(A>1\). b) M has Gaussian curvature \(K\leq -c^ 2<0\) and dimension 2. c) M is complete and simply connected with sectional curvature \(K\leq -c^ 2<0\) and M satisfies the convex conic neighbourhood condition at each point \(x\in M(\infty)\) (M is said to satisfy the convex conic neighborhood condition at a point \(x\in M(\infty)\) if for each point \(y\neq x\) in \(M(\infty)\) there exist disjoint open neighborhoods \(V_ x,V_ y\) of x,y in \(\bar M=M\cup M(\infty)\) such that \(V_ x\cap M\) is convex with \(C^ 2\) boundary). The author observes that under the hypothesis \(K\leq -c^ 2<0\) the convex conic neighborhood condition is satisfied if M is 2- dimensional or if M is rotationally symmetric, thereby giving another proof of b) and another proof of a) in a special case. M. T. Anderson later proved -[see the next review]- that the convex conic neighborhood condition is satisfied if the sectional curvature is bounded between two negative constants, thereby solving the asymptotic Dirichlet problem in this case. D. Sullivan also independently solved the asymptotic Dirichlet problem in the same case by using probabilistic methods [see the second review below]. With regard to condition a) above J. Milnor in [Am. Math. Mon. 84, 43-46 (1977; Zbl 0356.53002)] observed that the condition on the radial curvature implies in the case \(n=2\) that M has the complex structure of the unit disk. Moreover if one replaces A by 1 in the radial curvature expression and reverses the inequality then M has the structure of the complex plane and the asymptotic Dirichlet problem cannot be solved.
In the last part of the paper the author considers noncompact, nonsimply connected surfaces whose fundamental groups are finitely generated and whose ends are either ”standard expanding” or ”standard shrinking”. For such surfaces one may pose an analogous asymptotic Dirichlet problem for each expanding end, and the author shows that this problem is solvable. As an application he obtains the following theorem: Let M be a finitely connected, complete surface (not necessarily orientable) with curvature \(K\leq -c^ 2<0\). Assume that M has infinite total absolute curvature and is not topologically a Möbius band. Then M admits infinitely many nonconstant bounded harmonic functions.
Reviewer: P.Eberlein

MSC:

53C20 Global Riemannian geometry, including pinching
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31C12 Potential theory on Riemannian manifolds and other spaces
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