A natural differential operator on conic spaces

Pages: 568 - 577, Issue Special, September 2011

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Daniel Grieser - Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany (email)

Abstract: We introduce the notion of a conic space, as a natural structure on a manifold with boundary, and de ne a natural fi rst order di fferential operator, $c_d_\partial$, acting on boundary values of conic one-forms. Conic structures arise, for example, from resolutions of manifolds with conic singularities, embedded in a smooth ambient space. We show that pull-backs of smooth ambient one-forms to the resolution are cd@-closed, and that this is the only local condition on oneforms that is invariantly de ned on conic spaces. The operator $c_d_\partial$ extends to conic Riemannian metrics, and $c_d_\partial$-closed conic metrics have important geometric properties like the existence of an exponential map at the boundary.

Keywords:  conic singularity, rescaled tangent bundle, resolution
Mathematics Subject Classification:  Primary: 58J99

Received: July 2010;      Revised: January 2011;      Published: October 2011.