{"title":"黎曼流形上的协变薛定谔算子和 L2- 消失特性","authors":"","doi":"10.1016/j.difgeo.2024.102191","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>M</em> be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let <span><math><mi>E</mi></math></span> be a Hermitian vector bundle over <em>M</em> equipped with a metric covariant derivative ∇. We consider the operator <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>∇</mi><mo>+</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>V</mi></math></span>, where <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of <span><math><mi>E</mi></math></span>, <em>X</em> is a smooth (real) vector field on <em>M</em>, and <em>V</em> is a fiberwise self-adjoint, smooth section of the endomorphism bundle <span><math><mi>End</mi><mspace></mspace><mi>E</mi></math></span>. We give a sufficient condition for the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub></math></span>. As a corollary, putting <span><math><mi>X</mi><mo>≡</mo><mn>0</mn></math></span> and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>D</em> is the Dirac operator corresponding to ∇. In particular, when <span><math><mi>E</mi><mo>=</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the Hodge–deRham Laplacian on (complex-valued) <em>k</em>-forms, we recover some recent vanishing results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-harmonic (complex-valued) <em>k</em>-forms.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds\",\"authors\":\"\",\"doi\":\"10.1016/j.difgeo.2024.102191\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>M</em> be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let <span><math><mi>E</mi></math></span> be a Hermitian vector bundle over <em>M</em> equipped with a metric covariant derivative ∇. We consider the operator <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>∇</mi><mo>+</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>V</mi></math></span>, where <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of <span><math><mi>E</mi></math></span>, <em>X</em> is a smooth (real) vector field on <em>M</em>, and <em>V</em> is a fiberwise self-adjoint, smooth section of the endomorphism bundle <span><math><mi>End</mi><mspace></mspace><mi>E</mi></math></span>. We give a sufficient condition for the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub></math></span>. As a corollary, putting <span><math><mi>X</mi><mo>≡</mo><mn>0</mn></math></span> and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>D</em> is the Dirac operator corresponding to ∇. In particular, when <span><math><mi>E</mi><mo>=</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the Hodge–deRham Laplacian on (complex-valued) <em>k</em>-forms, we recover some recent vanishing results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-harmonic (complex-valued) <em>k</em>-forms.</p></div>\",\"PeriodicalId\":51010,\"journal\":{\"name\":\"Differential Geometry and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Geometry and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224524000846\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224524000846","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
假设 M 是满足加权波恩卡列不等式的完整黎曼流形,假设 E 是 M 上的赫尔墨斯向量束,并配有度量协变导数∇。我们考虑算子 HX,V=∇†∇+∇X+V,其中∇† 是∇关于 E 的平方可积分截面空间内积的形式邻接,X 是 M 上的光滑(实)向量场,V 是内形束 EndE 的纤维自交光滑截面。我们给出了 HX,V 的 L2 内核三性的充分条件。作为推论,假设 X≡0 并在配备了克利福德连接∇的克利福德模块的环境中工作,我们会得到 D2 的 L2 内核的三性,其中 D 是对应于∇的狄拉克算子。特别是,当 E=ΛCkT⁎M 和 D2 是(复值)k 形式上的霍奇-德拉姆拉普拉卡时,我们恢复了 L2 谐波(复值)k 形式的一些最新消失结果。
Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds
Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator , where is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of , X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle . We give a sufficient condition for the triviality of the -kernel of . As a corollary, putting and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the -kernel of , where D is the Dirac operator corresponding to ∇. In particular, when and is the Hodge–deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for -harmonic (complex-valued) k-forms.
期刊介绍:
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.