Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

Abstract

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L 2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.Mathematics Subject Classification(2000) : 53C42, 58C40.

Stable Minimal Hypersurfaces With Weighted Poincaré Inequality in a Riemannian Manifold

“…We show that we still get the vanishing property without assuming that the hypersurfaces is non-totally geodesic. This generalizes a result in [2].…”

“…In the other direction, Dung and Seo ( [2]) considered a complete stable minimal hypersurface in a Riemannian manifold with sectional curvature bounded below by a nonpositive constant and proved the following theorem.…”

Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold

“…In [32], Yun proved that if M ⊂ R n+1 is a complete minimal hypersurface with sufficiently small total scalar curvature M |A| n , then there is no nontrivial L 2 harmonic 1-form on M . Yun's result has been generalized into various ambient spaces [2,6,[23][24][25].…”

L2HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

“…If we choose ρ(x) = λ 1 (M ) in Theorem 1.2, we can get the following corollary which is an extension of Theorem 8 in [5] without the assumption of non-totally geodesic and minimality of a hypersurface, and dimension restriction arises in the estimating of the Ricci curvature.…”

“…Later, this result was generalized by Dung and Seo [5] to a complete stable minimal hypersurface in a Riemannnian manifold with sectional curvature bounded below by a nonpositive constant, and proved that complete noncompact stable non-totally geodesic minimal hypersurface in Riemannian manifold N with…”