文摘
Let lsi1" class="mathmlsrc">lass="formulatext 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">Mnlass="mathContainer hidden">lass="mathCode"> be a complete non-compact submanifold in the hyperbolic space lsi2" class="mathmlsrc">lass="formulatext 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">Hn+mlass="mathContainer hidden">lass="mathCode">. We first give an estimate for the bottom of the spectral of the Laplace operator on lsi1" class="mathmlsrc">lass="formulatext 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">Mnlass="mathContainer hidden">lass="mathCode">, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for lsi4" class="mathmlsrc">lass="formulatext 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">L2lass="mathContainer hidden">lass="mathCode"> harmonic forms in certain degrees if the total mean curvature of lsi1" class="mathmlsrc">lass="formulatext 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">Mnlass="mathContainer hidden">lass="mathCode"> 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 lsi4" class="mathmlsrc">lass="formulatext 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">L2lass="mathContainer hidden">lass="mathCode"> harmonic lsi7" class="mathmlsrc">lass="formulatext 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">1lass="mathContainer hidden">lass="mathCode">-forms, we can further prove some one-end theorems.