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Resolvent estimates and local decay of waves on conic manifolds. (English) Zbl 1296.53075

Summary: We consider manifolds with conic singularities that are isometric to \(\mathbb{R}^n\) outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process.
The proof of the resolvent estimate relies on the propagation of singularities theorems of R. Melrose and the second author [Invent. Math. 156, No. 2, 235–299 (2004; Zbl 1088.58011)] to establish a “very weak” Huygens’ principle, which may be of independent interest.
As applications of the estimate, we obtain an exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schrödinger equation.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 1088.58011
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