Biharmonic hypersurfaces with bounded mean curvature

• Maeta, Shun

抄録

We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon left-parenthesis upper M Superscript m Baseline comma g right-parenthesis right-arrow left-parenthesis upper S Superscript m plus 1 Baseline comma h right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mo>:</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :(M^m,g)\rightarrow (S^{m+1},h)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a sphere. If the squared norm of the second fundamental form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded from above by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper M Endscripts upper H Superscript negative p Baseline d v Subscript g Baseline greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>M</mml:mi> </mml:msub> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _M H^{- p }dv_g>\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0>p>\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the mean curvature is constant. This is an affirmative partial answer to the BMO conjecture for biharmonic submanifolds.

収録刊行物

• Proceedings of the American Mathematical Society

  Proceedings of the American Mathematical Society 145 (4), 1773-1779, 2017-04

  American Mathematical Society

キーワード

• Applied Mathematics
• General Mathematics

詳細情報

• CRID: 1050282813462938752
• NII論文ID: 120006352343
• NII書誌ID: AA1245450X
• ISSN: 10886826
• ISSN: 00029939
• DOI: 10.1090/proc/13335
• Web Site: http://ir.lib.shimane-u.ac.jp/40623
• Web Site: https://www.ams.org/proc/2017-145-04/S0002-9939-2016-13335-2/proc13335_AM.pdf
• Web Site: http://www.ams.org/proc/2017-145-04/S0002-9939-2016-13335-2/S0002-9939-2016-13335-2.pdf
• Web Site: https://www.ams.org/proc/2017-145-04/S0002-9939-2016-13335-2/S0002-9939-2016-13335-2.pdf
• 本文言語コード: en
• 資料種別: journal article
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