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The main purpose of this short note is to derive some generalizations of theclassical long neck principle and give a spectral width inequality of geodesiccollar neighborhood. Our results are obtained via the spinorial Calliasoperator approach. An important step is to introduce the relative Gromov-Lawsonpair on a compact manifold with boundary, relative to a background manifold.

A Note on the Long Neck Principle and Spectral Width Inequality of Geodesic Collar Neighborhood