A new characterization of submanifolds with parallel mean curvature vector in <I>S</I><SUP><I>n</I>+<I>p</I></SUP>
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概要
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In this work we will consider compact submanifold <I>M</I><SUP><I>n</I></SUP> immersed in the Euclidean sphere <I>S</I><SUP><I>n</I>+<I>p</I></SUP> with parallel mean curvature vector and we introduce a Schrödinger operator <I>L</I>=−Δ+<I>V</I>, where Δ stands for the Laplacian whereas <I>V</I> is some potential on <I>M</I><SUP><I>n</I></SUP> which depends on <I>n</I>, <I>p</I> and <I>h</I> that are respectively, the dimension, codimension and mean curvature vector of <I>M</I><SUP><I>n</I></SUP>. We will present a gap estimate for the first eigenvalue μ<SUB>1</SUB> of <I>L</I>, by showing that either μ<SUB>1</SUB>=0 or μ<SUB>1</SUB>≤−<I>n</I>(1+<I>H</I><SUP>2</SUP>). As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds.
- 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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