Minimal submanifolds with small total scalar curvature in Euclidean space

Journal of Mathematical Analysis and Applications

2010

Self Cite

Section: Introductionmentioning

confidence: 92%

L2 harmonic 1-forms on minimal submanifolds in hyperbolic space

Journal of Mathematical Analysis and Applications

2010

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“…While the L 2 theory is quite well understood, in the case p = 2, the L p theory is less developed. See [Scott 1995] for general L p theory of differential forms on a manifold.…”

Section: Introductionmentioning

confidence: 99%

L^pharmonic 1-forms and first eigenvalue of a stable minimal hypersurface

2014

Pacific J. Math.

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We estimate the bottom of the spectrum of the Laplace operator on a stable minimal hypersurface in a negatively curved manifold. We also derive various vanishing theorems for L p harmonic 1-forms on minimal hypersurfaces in terms of the bottom of the spectrum of the Laplace operator. As consequences, the corresponding Liouville type theorems for harmonic functions with finite L p energy on minimal hypersurfaces in a Riemannian manifold are obtained.

“…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].…”

Section: Introductionmentioning

confidence: 96%

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

Dũng

Journal of Mathematical Analysis and Applications

2015

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Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6, we prove that if M satisfies the δ-stability inequality (0 < δ ≤ 1), then there is no nontrivial L 2β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L 2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature K N satisfying that −k 2 ≤ K N ≤ 0 for some constant k.