Eigenvalue estimates without Bakry-Emery-Ricci bounds

2019/01/17 by Gabriel Khan, Khan, Gabriel
Mathematics · #58J05 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1901.06277

openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an L^∞-drift term. We make no assumptions that the operator is self-adjoint or that the drift has any additional regularity. In the case where the operator is self-adjoint, this establishes a lower bound on the spectrum without assuming a lower bound for the Bakry-Emery Ricci tensor. Put colloquially, this result states that no matter which way the wind blows, heat will diffuse at a definite rate depending only on the geometry of the underlying space and the maximal wind speed.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the parabolic kernel of the Schrödinger operator
• Can One Hear the Shape of a Drum?

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