Abstract
AbstractIn this paper, we investigate the rigidity and vanishing properties of compact submanifolds with free boundary of arbitrary codimension in the unit ball. We first show that a minimal submanifold with free boundary in the unit ball satisfying a pointwise or integral curvature pinching condition on the second fundamental form is a flat equatorial disk. Then we prove a vanishing theorem for cohomology groups for submanifolds with free boundary in the unit ball under an integral curvature pinching condition on the trace‐free second fundamental form.
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