Extrinsic GJMS operators for submanifolds

  • Jeffrey S. Case

    Penn State University, University Park, USA
  • C Robin Graham

    University of Washington, Seattle, USA
  • Tzu-Mo Kuo

    National Center for Theoretical Sciences, Taipei, Taiwan
Extrinsic GJMS operators for submanifolds cover
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Abstract

We derive extrinsic GJMS operators and -curvatures associated to a submanifold of a conformal manifold. The operators are conformally covariant scalar differential operators on the submanifold with leading part a power of the Laplacian in the induced metric. Upon realizing the conformal manifold as the conformal infinity of an asymptotically Poincaré–Einstein space and the submanifold as the boundary of an asymptotically minimal submanifold thereof, these operators arise as obstructions to smooth extension as eigenfunctions of the Laplacian of the induced metric on the minimal submanifold. We derive explicit formulas for the operators of orders 2 and 4. We prove factorization formulas when the original submanifold is a minimal submanifold of an Einstein manifold. We also show how to reformulate the construction in terms of the ambient metric for the conformal manifold, and use this to prove that the operators defined by the factorization formulas are conformally invariant for all orders in all dimensions.

Cite this article

Jeffrey S. Case, C Robin Graham, Tzu-Mo Kuo, Extrinsic GJMS operators for submanifolds. Rev. Mat. Iberoam. (2025), published online first

DOI 10.4171/RMI/1545