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Abstract:
We consider how the geometry and topology of a compact n-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich, and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In dimension two, we show that a flat disk is not only Steklov isospectral to a cone but, in fact, a disk and cone of appropriate size have identical Dirichlet-to-Neumann operators. This provides a counterexample to the inverse tomography problem in the orbifold setting and contrasts with results of Lassas and Uhlmann in the manifold setting. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi, and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are
solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.
