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Explore a 52-minute seminar from the "Spectral Geometry in the clouds" series, delivered by Changwei Xiong from Sichuan University. Delve into the establishment of a weighted Reilly formula for differential forms on compact Riemannian manifolds with boundaries. Discover applications of this formula, including a sharp lower bound for the first positive eigenvalue of the Steklov eigenvalue problem on differential forms. Examine a comparison result between the spectrum of this Steklov eigenvalue problem and the spectrum of the Hodge Laplacian on the manifold's boundary. Conclude by discussing an open problem for differential forms analogous to Escobar's conjecture for functions. This talk, based on the preprint arXiv:2312.16780, offers a deep dive into advanced topics in spectral geometry and differential forms.
