Upper bounds for the first eigenvalue of the Dirac operator on surfaces

1998/06/15 by Ilka Agricola, Thomas Friedrich · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DG #msc:53A05 #msc:58G25

paper · pdf · doi:10.1016/s0393-0440(98)00032-1

Latex2.09, 23 pages

arxiv created 1998/06/15 · openalex publication_date 1999/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface M² \hookrightarrow \Bbb R³ as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics.

Citations

• On the spinor representation of surfaces in Euclidean 3-space
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Cited by

• A geometric estimate for a periodic Schrödinger operator whose potential is the curvature of a spherical curve