Conformal Structures and Rigidity of Complete Stable Minimal Hypersurfaces

초록

We study complete stable minimal hypersurfaces in Riemannian manifolds under various curvature assumptions. In an (n+1) -dimensional complete oriented manifold with nonnegative scalar curvature, we prove that no complete oriented noncompact stable minimal hypersurface can be conformally equivalent to a bounded domain in an n-dimensional manifold with nonpositive scalar curvature. As a consequence, there exists no complete stable minimal hypersurface in Rn+1 that is conformally equivalent to a bounded domain in Rn . We also show that compact stable minimal hypersurfaces in manifolds with nonnegative scalar curvature have nonnegative smooth Yamabe invariant and we characterize the equality case. Next, we consider complete two-sided stable minimal hypersurfaces in ambient manifolds with pinched sectional curvature. Under an additional curvature condition that makes the second fundamental form a Codazzi tensor, we obtain a rigidity result under an L2 -condition on the second fundamental form and derive an upper bound for the first eigenvalue of the Laplacian. Finally, we establish that any complete noncompact two-sided minimal hypersurface immersed in a warped product manifold is stable, provided that the angle function is positive and the second derivative of the warping function is nonnegative.

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