Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space

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• 作者：Hezi Lin ; ^{lhz1@fjnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C21 ; 53C42
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：126-137
• 全文大小：636 K

文摘

Let mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=6dccbfac4a0ab271a6b99fd47e600b08" title="Click to view the MathML source">MⁿmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi>Mmi>mrow><mrow><mi>nmi>mrow>msup>math> be a complete non-compact submanifold in the hyperbolic space mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si2.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=898021c9aafde6aefc17213c5c6e65d0" title="Click to view the MathML source">H^n+mmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msup><mrow><mi mathvariant="double-struck">Hmi>mrow><mrow><mi>nmi><mo>+mo><mi>mmi>mrow>msup>math>. We first give an estimate for the bottom of the spectral of the Laplace operator on mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=6dccbfac4a0ab271a6b99fd47e600b08" title="Click to view the MathML source">MⁿmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi>Mmi>mrow><mrow><mi>nmi>mrow>msup>math>, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=63f59013af68bef4f0378949d2e2b32f" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msup><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msup>math> harmonic forms in certain degrees if the total mean curvature of mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=6dccbfac4a0ab271a6b99fd47e600b08" title="Click to view the MathML source">MⁿmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi>Mmi>mrow><mrow><mi>nmi>mrow>msup>math> is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=63f59013af68bef4f0378949d2e2b32f" title="Click to view the MathML source">L²mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msup><mrow><mi>Lmi>mrow><mrow><mn>2mn>mrow>msup>math> harmonic mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302243&_mathId=si7.gif&_user=111111111&_pii=S0362546X16302243&_rdoc=1&_issn=0362546X&md5=65b5ed1497190a32a72c61f2ab348267" title="Click to view the MathML source">1mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mn>1mn>math>-forms, we can further prove some one-end theorems.