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Dirac operator on embedded hypersurfaces. (English) Zbl 0988.53019

The present paper is a continuation of a previous work by the authors [Math. Res. Lett. 8, No. 1-2, 195-208 (2001; Zbl 0988.53019)]. They present new extrinsic lower bounds for the classical Dirac operator on the boundary of a compact domain of a spin manifold by solving some boundary problems using boundary conditions of the Atiyah-Patodi-Singer type. Finally, spinorial techniques are employed to obtain simple proofs of various classical results for compact embedded hypersurfaces. Contents include: an introduction and preliminaries; bounding domains hypersurfaces; boundary problems for the Dirac operator; extrinsic lower bounds for the hypersurface Dirac operator; constant mean curvature and minimal embedded hypersurfaces.

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C40 Global submanifolds
53C80 Applications of global differential geometry to the sciences
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0988.53019
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