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1. Spectral Gaps of the Laplacian on Differential Forms

   Lu, Zhiqin; Leal, Healton (January 2022, Contemporary mathematics)

   In this short article, we explore some basic results associated to the Generalized Weyl criterion for the essential spectrum of the Laplacian on Riemannian manifolds. We use the language of Gromov-Hausdorff convergence to recall a spectral gap theorem. Finally, we make the necessary adjustments to extend our main results, and construct a class of complete noncompact manifolds with an arbitrarily large number of gaps in the spectrum of the Hodge Laplacian acting on differential forms. 

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2. Analysis of the Laplacian on the moduli space of poliarzied Calabi-Yau manifolds

   Lu, Zhiqin Xu (January 2019, Contemporary mathematics)

   In this paper, we generalize the spectrum relation in the paper "On the spectrum of the Laplacian, Math. Ann., 359(1-2):211--238, 2014, by Nelia Charalambous and Zhiqin Lu" to any Hermitian manifolds. We also prove that the closure of the Laplace operator on the moduli space of polarized Calabi-Yau manifolds is self-adjoint. 

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3. Sarnak’s spectral gap question

   Kelmer, Dubi; Kontorovich, Alex; Lutsko, Christopher (December 2023, Journal d'Analyse Mathématique)

   We answer in the affirmative a question of Sarnak’s from 2007, confirming that the Patterson–Sullivan base eigenfunction is the unique square-integrable eigenfunction of the hyperbolic Laplacian invariant under the group of symmetries of the Apollonian packing. Thus the latter has a maximal spectral gap. We prove further restrictions on the spectrum of the Laplacian on a wide class of manifolds coming from Kleinian sphere packings. 

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4. Graphs with positive spectrum

   https://doi.org/10.1112/jlms.12733  

   Hua, Bobo; Lu, Zhiqin (March 2023, Journal of the London Mathematical Society)

   Abstract In this paper, we prove sharp decay estimates of nonnegative generalized subharmonic functions on graphs with positive Laplacian spectrum, which extends the result by Li and Wang (J. Differential Geom. 58 (2001) 501–534) on Riemannian manifolds. 

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5. Sobolev and Schatten estimates for the complex Green operator on spheres

   Kim, Elena; Ogden, W. Jacob; Reerink, Tommie; Zeytuncu, Yunus E. (February 2020, New York journal of mathematics)

   The complex Green operator $$\mathcal{G}$$ on CR manifolds is the inverse of the Kohn-Laplacian $$\square_b$$ on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for $$\mathcal{G}$$ on the unit sphere $$\mathbb{S}^{2n-1}\subset \mathbb{C}^n$$. We obtain these estimates by using the spectrum of $$\boxb$$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator. 

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