Area spectral rigidity for axially symmetric and Radon domains

2024/10/16 by Baracco, Luca, Bernardi, Olga, Nardi, Alessandra
#37C83 #58J53 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.12644

Abstract

We prove that (a) any finitely smooth axially symmetric strictly convex domain, with everywhere positive curvature and sufficiently close to an ellipse, and (b) any finitely smooth centrally symmetric strictly convex domain, even-rationally integrable, with everywhere positive curvature and sufficiently close to an ellipse is area spectrally rigid. This means that any area-isospectral family of domains in these classes is necessarily equi-affine. We use the same technique -- adapted to symplectic billiards -- of the paper by De Simoi, Kaloshin, and Wei (2017). The novelty is that the result holds for axially symmetric domains, as in the cited paper, as well as for a subset of centrally symmetric ones.

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