Geometry of the Smallest 1-form Laplacian Eigenvalue on Hyperbolic Manifolds
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2018-12-01
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Abstract
We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth in families of hyperbolic 3-manifolds Benjamini–Schramm converging to H3.
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Lipnowski, M, and M Stern (2018). Geometry of the Smallest 1-form Laplacian Eigenvalue on Hyperbolic Manifolds. Geometric and Functional Analysis, 28(6). pp. 1717–1755. 10.1007/s00039-018-0471-x Retrieved from https://hdl.handle.net/10161/24072.
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Scholars@Duke
Mark A. Stern
The focus of Professor Stern's research is the study of analytic problems arising in geometry and physics.
In recent and ongoing work, Professor Stern has studied analytical, geometric, and topological questions arising in Yang-Mills theory. These include analyzing the moduli space of Yang Mills instantons on gravitational instantons, analyzing the asymptotic structure of instantons (proving a nonlinear analog of the inverse square law of electromagnetism), and analyzing the structure of singularities of instantons and of harmonic maps.
In addition, Professor Stern has recently studied questions arising in the interplay between geometric group theory and Lp and L2 cohomology. This work includes finding new bounds on L2 betti numbers of negatively curved manifolds, and new growth,
stability, and vanishing results for Lp and L2 cohomology of symmetric and locally symmetric spaces.
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