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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.00926 |
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| _version_ | 1866929478428524544 |
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| author | Milatovic, Ognjen |
| author_facet | Milatovic, Ognjen |
| contents | Let $M$ be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $\mathcal{E}$ be a Hermitian vector bundle over $M$ equipped with a metric covariant derivative $\nabla$. We consider the operator $H_{X,V}=\nabla^{\dagger}\nabla+\nabla_{X}+ V$, where $\nabla^{\dagger}$ is the formal adjoint of $\nabla$ with respect to the inner product in the space of square-integrable sections of $\mathcal{E}$, $X$ is a smooth (real) vector field on $M$, and $V$ is a fiberwise self-adjoint, smooth section of the endomorphism bundle $\textrm{End }\mathcal{E}$. We give a sufficient condition for the triviality of the $L^2$-kernel of $H_{X,V}$. As a corollary, putting $X\equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection $\nabla$, we obtain the triviality of the $L^2$-kernel of $D^2$, where $D$ is the Dirac operator corresponding to $\nabla$. In particular, when $\mathcal{E}=Λ_{\mathbb{C}}^{k}T^*M$ and $D^2$ is the Hodge--deRham Laplacian on (complex-valued) $k$-forms, we recover some recent vanishing results for $L^2$-harmonic (complex-valued) $k$-forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Covariant Schrödinger Operator and $L^2$-Vanishing Property on Riemannian Manifolds Milatovic, Ognjen Differential Geometry Let $M$ be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $\mathcal{E}$ be a Hermitian vector bundle over $M$ equipped with a metric covariant derivative $\nabla$. We consider the operator $H_{X,V}=\nabla^{\dagger}\nabla+\nabla_{X}+ V$, where $\nabla^{\dagger}$ is the formal adjoint of $\nabla$ with respect to the inner product in the space of square-integrable sections of $\mathcal{E}$, $X$ is a smooth (real) vector field on $M$, and $V$ is a fiberwise self-adjoint, smooth section of the endomorphism bundle $\textrm{End }\mathcal{E}$. We give a sufficient condition for the triviality of the $L^2$-kernel of $H_{X,V}$. As a corollary, putting $X\equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection $\nabla$, we obtain the triviality of the $L^2$-kernel of $D^2$, where $D$ is the Dirac operator corresponding to $\nabla$. In particular, when $\mathcal{E}=Λ_{\mathbb{C}}^{k}T^*M$ and $D^2$ is the Hodge--deRham Laplacian on (complex-valued) $k$-forms, we recover some recent vanishing results for $L^2$-harmonic (complex-valued) $k$-forms. |
| title | Covariant Schrödinger Operator and $L^2$-Vanishing Property on Riemannian Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2405.00926 |