The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues
In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and or...
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2024-12-01
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| author | Josef Mikeš Sergey Stepanov Irina Tsyganok |
| author_facet | Josef Mikeš Sergey Stepanov Irina Tsyganok |
| author_sort | Josef Mikeš |
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| description | In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian. |
| format | Article |
| id | doaj-art-f59b1d3628f34e4bbdcaaddfaf102d64 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-f59b1d3628f34e4bbdcaaddfaf102d642025-08-20T02:56:52ZengMDPI AGMathematics2227-73902024-12-011224393610.3390/math12243936The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest EigenvaluesJosef Mikeš0Sergey Stepanov1Irina Tsyganok2Department of Algebra and Geometry, Palacký University Olomouc, 771 47 Olomouc, Czech RepublicDepartment of Mathematics, Finance University, 125468 Moscow, RussiaDepartment of Mathematics, Finance University, 125468 Moscow, RussiaIn the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian.https://www.mdpi.com/2227-7390/12/24/3936Lichnerowicz Laplaciansymmetric tensorsexterior differential formvanishing theoremeigenvalue |
| spellingShingle | Josef Mikeš Sergey Stepanov Irina Tsyganok The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues Mathematics Lichnerowicz Laplacian symmetric tensors exterior differential form vanishing theorem eigenvalue |
| title | The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues |
| title_full | The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues |
| title_fullStr | The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues |
| title_full_unstemmed | The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues |
| title_short | The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues |
| title_sort | lichnerowicz type laplacians vanishing theorems for their kernels and estimate theorems for their smallest eigenvalues |
| topic | Lichnerowicz Laplacian symmetric tensors exterior differential form vanishing theorem eigenvalue |
| url | https://www.mdpi.com/2227-7390/12/24/3936 |
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