2015/02/09 by Jeffrey S. Case, Sun‐Yung Alice Chang, Sun‐Yung A. Chang · 2 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research
paper · doi:10.1002/cpa.21564
We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet‐to‐Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli‐Silvestre extension for (−Δ) γ when γ ∊ (0,1), and both a geometric interpretation and a curved analogue of the higher‐order extension found by R. Yang for (−Δ) γ when γ > 1 . We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré‐Einstein manifold, including an interpretation as a renormalized energy. Second, for γ ∊ (1,2), we show that if the scalar curvature and the fractional Q ‐curvature Q 2γ of the boundary are nonnegative, then the fractional GJMS operator P 2γ is nonnegative. Third, by assuming additionally that Q 2γ is not identically zero, we show that P 2γ satisfies a strong maximum principle.© 2016 Wiley Periodicals, Inc.