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An Alternative for Constant Mean Curvature Hypersurfaces
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abstract
Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$ equipped with a generic Riemannian metric $g$. Let $c$ be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean curvature equal to $c$, or there exist infinitely many distinct closed hypersurfaces with constant mean curvature less than $c$ but enclosing half the volume of $M$.
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Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface
On any closed 3-7 dimensional manifold that contains a strictly stable minimal surface, a large class of prescribed-mean-curvature functions admit infinitely many distinct almost embedded hypersurfaces.
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