Asymptotic dirichlet problem for schrodinger operator and rough isometry

  • Yoon, Jaihan (Department of Mathematics, Seoul National University)
  • Published : 1997.02.01


The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)


  1. J. Diff. Geom. v.18 The Dirichlet problem at infinity for manifolds of negative curvature M. T. Anderson
  2. Ann. Math. v.121 Positive harmonic functions on complete manifolds of negative curvature M. T. Anderson;R. Schoen
  3. Ann. Sci. Ec. Norm. Sup. Paris v.15 A note on the isoperimetric constant P. Buser
  4. Comm. Anal. Geom. v.1 The Dirichlet problem at infinity for nonpositively curved manifolds S. Y. Cheng
  5. Trans. of Amer. Math. Soc. v.281 Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds H. I. Choi
  6. Rough Isometry and the Asymptotic Dirichlet Problem H. I. Choi;S. W. Kim;Y. H. Lee
  7. Varietes riemanniennes isometriques alinfini Th. Coulhon;L. Saloff-Coste
  8. On harmonic rough-isometries P. Li;J. Wang
  9. Springer Lecture Notes in Mathmatics v.1201 Analytic inequalities, and rough isometries between noncompact Riemannian manifolds M. Kanai
  10. J. Math. Soc. Japan v.37 Rough isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds M. Kanai
  11. J. Math. Soc. Japan v.38 Rough isometries and the parabolicity of riemannian manifolds M. Kanai
  12. Lectures on Differential Geometry R. Schoen;S. T. Yau
  13. J. Diff. Geom. v.18 The Dirichlet problem at infinity for a negatively curved manifold D. Sullivan