New Results About the Lambda Constant and Ground States of the 𝑊-Functional
In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of gro...
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| Format: | Article |
| Language: | English |
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De Gruyter
2020-08-01
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| Series: | Advanced Nonlinear Studies |
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| Online Access: | https://doi.org/10.1515/ans-2020-2077 |
| _version_ | 1811280119941038080 |
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| author | Ma Li |
| author_facet | Ma Li |
| author_sort | Ma Li |
| collection | DOAJ |
| description | In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold (M,g){(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold (M,g){(M,g)}, if the scalar curvature s of (M,g){(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s=0{s=0} in M. We also introduce a new constant d(M,g){d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method. |
| first_indexed | 2024-04-13T01:07:44Z |
| format | Article |
| id | doaj.art-7ef77bbca0e14ee88f0179957b1752f2 |
| institution | Directory Open Access Journal |
| issn | 1536-1365 2169-0375 |
| language | English |
| last_indexed | 2024-04-13T01:07:44Z |
| publishDate | 2020-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-7ef77bbca0e14ee88f0179957b1752f22022-12-22T03:09:16ZengDe GruyterAdvanced Nonlinear Studies1536-13652169-03752020-08-0120365166110.1515/ans-2020-2077New Results About the Lambda Constant and Ground States of the 𝑊-FunctionalMa Li0School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian DistrictBeijing, 100083; and Department of Mathematics, Henan Normal University, Xinxiang, 453007P. R. ChinaIn this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold (M,g){(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold (M,g){(M,g)}, if the scalar curvature s of (M,g){(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s=0{s=0} in M. We also introduce a new constant d(M,g){d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method.https://doi.org/10.1515/ans-2020-2077lambda constantsground stateslions’ concentration-compactness method53c2035b65 58j50 |
| spellingShingle | Ma Li New Results About the Lambda Constant and Ground States of the 𝑊-Functional Advanced Nonlinear Studies lambda constants ground states lions’ concentration-compactness method 53c20 35b65 58j50 |
| title | New Results About the Lambda Constant and Ground States of the 𝑊-Functional |
| title_full | New Results About the Lambda Constant and Ground States of the 𝑊-Functional |
| title_fullStr | New Results About the Lambda Constant and Ground States of the 𝑊-Functional |
| title_full_unstemmed | New Results About the Lambda Constant and Ground States of the 𝑊-Functional |
| title_short | New Results About the Lambda Constant and Ground States of the 𝑊-Functional |
| title_sort | new results about the lambda constant and ground states of the 𝑊 functional |
| topic | lambda constants ground states lions’ concentration-compactness method 53c20 35b65 58j50 |
| url | https://doi.org/10.1515/ans-2020-2077 |
| work_keys_str_mv | AT mali newresultsaboutthelambdaconstantandgroundstatesofthewfunctional |