Eigenvalue inequalities and minimal submanifolds
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概要
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Let (<I>S</I><SUP><I>m</I></SUP>, <I>g</I><SUB>0</SUB>) be the unit sphere, (<I>M</I><SUP><I>n</I></SUP>, <I>g</I>) its submanifold, λ<SUB>1</SUB> the first nonzero eigenvalue of (<I>M</I><SUP><I>n</I></SUP>, <I>g</I>), <I>H</I> the mean curvature vector field of <I>M</I><SUP><I>n</I></SUP>. By Takahashi theorem, if <I>M</I><SUP><I>n</I></SUP> is minimal, then λ<SUB>1</SUB>{≤}<I>n</I>. In this paper, we establish some eigenvalue inequalities and use them to prove:<BR>1. If <I>x</I> is mass symmetric and of order {<I>k</I>, <I>k</I>+1} for some <I>k</I> such that λ<SUB><I>k</I></SUB>{≥}<I>n</I> or λ<SUB><I>k</I>+1</SUB>{≤}<I>n</I>, then φ is minimal and λ<SUB><I>k</I></SUB>=<I>n</I> or λ<SUB><I>k</I>+1</SUB>=<I>n</I>.<BR>2. If <I>H</I> is parallel, ∫<SUB><I>M</I></SUB><I>Hdv</I><SUB><I>M</I></SUB>=0 and σ<SUP>2</SUP>{≤}λ<SUB>1</SUB>, then <I>H</I>=0 or σ<SUP>2</SUP>=λ<SUB>1</SUB>.<BR>3. If <I>H</I> is parallel and λ<SUB><I>k</I></SUB>=<I>n</I> for some <I>k</I>, then <I>H</I>=0 or σ<SUP>2</SUP>(<I>x</I>){≥}λ<SUB><I>k</I>+1</SUB>−λ<SUB><I>k</I></SUB> for some <I>x</I>∈<I>M</I><SUP><I>n</I></SUP>.<BR>4. λ<SUB>1</SUB>{≤}\frac{<I>nV</I><SUP>2</SUP>}{<I>V</I><SUP>2</SUP>−(∫<SUB><I>M</I></SUB><I>Hdv</I><SUB><I>M</I></SUB>)<SUP>2</SUP>}. Especially, if ∫<SUB><I>M</I></SUB><I>Hdv</I><SUB><I>M</I></SUB>=0, then λ<SUB>1</SUB>{≤}<I>n</I>, and that λ<SUB>1</SUB>=<I>n</I> implies that φ is minimal.
- 国立大学法人 東京工業大学大学院理工学研究科数学専攻の論文
国立大学法人 東京工業大学大学院理工学研究科数学専攻 | 論文
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