Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space
Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 h...
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| Veröffentlicht in: | Nonlinear analysis 2017-01, Vol.148, p.126-137 |
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| creator | Lin, Hezi |
| description | Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn 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 L2 harmonic 1-forms, we can further prove some one-end theorems. |
| doi_str_mv | 10.1016/j.na.2016.09.015 |
| format | Article |
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We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn 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 L2 harmonic 1-forms, we can further prove some one-end theorems.</description><identifier>ISSN: 0362-546X</identifier><identifier>EISSN: 1873-5215</identifier><identifier>DOI: 10.1016/j.na.2016.09.015</identifier><language>eng</language><publisher>Elmsford: Elsevier Ltd</publisher><subject>[formula omitted]-harmonic [formula omitted]-forms ; Complete submanifolds ; Curvature ; Ends ; First eigenvalue ; Gap theorems ; Harmonic analysis ; Manifolds (mathematics) ; Operators (mathematics) ; Theorems ; Topological manifolds</subject><ispartof>Nonlinear analysis, 2017-01, Vol.148, p.126-137</ispartof><rights>2016 Elsevier Ltd</rights><rights>Copyright Elsevier BV Jan 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c322t-1b9343c410f914403a1eb4832f6889cb8755b6b380a2365727e61f8c7ae44a1d3</citedby><cites>FETCH-LOGICAL-c322t-1b9343c410f914403a1eb4832f6889cb8755b6b380a2365727e61f8c7ae44a1d3</cites><orcidid>0000-0002-1072-2889</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0362546X16302243$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids></links><search><creatorcontrib>Lin, Hezi</creatorcontrib><title>Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space</title><title>Nonlinear analysis</title><description>Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn 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 L2 harmonic 1-forms, we can further prove some one-end theorems.</description><subject>[formula omitted]-harmonic [formula omitted]-forms</subject><subject>Complete submanifolds</subject><subject>Curvature</subject><subject>Ends</subject><subject>First eigenvalue</subject><subject>Gap theorems</subject><subject>Harmonic analysis</subject><subject>Manifolds (mathematics)</subject><subject>Operators (mathematics)</subject><subject>Theorems</subject><subject>Topological manifolds</subject><issn>0362-546X</issn><issn>1873-5215</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kMtLxDAQxoMouD7uHgOeWzPNo603WdYHrHhR8BbSdLqb0m1q0i7sf2-X9eppBub7Zr75EXIHLAUG6qFNe5Nmc5eyMmUgz8gCipwnMgN5ThaMqyyRQn1fkqsYW8YY5FwtyPvKbbDfm25CinF0OzMiNX1NN2ag4xZ9wF2kjQ80TtXO9K7xXR2p649Duj0MGCrfOUvjYCzekIvGdBFv_-o1-XpefS5fk_XHy9vyaZ1YnmVjAlXJBbcCWFOCEIwbwEoUPGtUUZS2KnIpK1XxgpmMK5lnOSpoCpsbFMJAza_J_WnvEPzPNOfWrZ9CP5_UUArIGZNQzip2UtngYwzY6CHMD4aDBqaP0HSre6OP0DQr9QxttjyeLDin3zsMOlqHvcXaBbSjrr373_wLmx1y0w</recordid><startdate>201701</startdate><enddate>201701</enddate><creator>Lin, Hezi</creator><general>Elsevier Ltd</general><general>Elsevier BV</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0002-1072-2889</orcidid></search><sort><creationdate>201701</creationdate><title>Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space</title><author>Lin, Hezi</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c322t-1b9343c410f914403a1eb4832f6889cb8755b6b380a2365727e61f8c7ae44a1d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>[formula omitted]-harmonic [formula omitted]-forms</topic><topic>Complete submanifolds</topic><topic>Curvature</topic><topic>Ends</topic><topic>First eigenvalue</topic><topic>Gap theorems</topic><topic>Harmonic analysis</topic><topic>Manifolds (mathematics)</topic><topic>Operators (mathematics)</topic><topic>Theorems</topic><topic>Topological manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lin, Hezi</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Nonlinear analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lin, Hezi</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space</atitle><jtitle>Nonlinear analysis</jtitle><date>2017-01</date><risdate>2017</risdate><volume>148</volume><spage>126</spage><epage>137</epage><pages>126-137</pages><issn>0362-546X</issn><eissn>1873-5215</eissn><abstract>Let Mn be a complete non-compact submanifold in the hyperbolic space Hn+m. We first give an estimate for the bottom of the spectral of the Laplace operator on Mn, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L2 harmonic forms in certain degrees if the total mean curvature of Mn 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 L2 harmonic 1-forms, we can further prove some one-end theorems.</abstract><cop>Elmsford</cop><pub>Elsevier Ltd</pub><doi>10.1016/j.na.2016.09.015</doi><tpages>12</tpages><orcidid>https://orcid.org/0000-0002-1072-2889</orcidid></addata></record> |
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| language | eng |
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| source | Elsevier ScienceDirect Journals |
| subjects | [formula omitted]-harmonic [formula omitted]-forms Complete submanifolds Curvature Ends First eigenvalue Gap theorems Harmonic analysis Manifolds (mathematics) Operators (mathematics) Theorems Topological manifolds |
| title | Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space |
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