{"title":"具有几何应用的非紧流形上广义调和形式的增长估计","authors":"S. Wei","doi":"10.1090/conm/756/15215","DOIUrl":null,"url":null,"abstract":"We introduce Condition W $\\,$(1.2) for a smooth differential form $\\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\\omega$ is a harmonic form on $M$ if and only if $\\omega$ is both closed and co-closed on $M\\, ,$ where $\\omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(\\ne 2) < 3\\, $ with $\\omega$ satisfying Condition W $\\,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1<q(\\ne 2)<3\\, $ satisfying Condition W $\\,$(1.2) is both closed and co-closed (cf. Theorem 1.1). This generalizes the work of A. Andreotti and E. Vesentini [AV] for every $L^2$ harmonic form $\\omega\\, .$ In extending $\\omega$ in $L^2$ to $L^q$, for $q \\ne 2$, Condition W $\\,$(1.2) has to be imposed due to counter-examples of D. Alexandru-Rugina$\\big($ [AR] p. 81, Remarque 3$\\big).$ We then study nonlinear partial differential inequalities for differential forms $ \\langle\\omega, \\Delta \\omega\\rangle \\ge 0, $ in which solutions $\\omega$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $\\omega\\, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $\\langle\\omega, \\Delta \\omega\\rangle \\ge 0\\, ,$ (ii) $\\Delta \\omega = 0\\, ,$ $($iii$)$$\\quad d\\, \\omega = d^{\\star}\\omega = 0\\, ,$ (iv) $\\langle \\star\\, \\omega, \\Delta \\star\\, \\omega\\rangle \\ge 0\\, ,$ (v) $\\Delta \\star\\, \\omega = 0\\, ,$ and (vi) $d\\, \\star\\, \\omega = d^{\\star} \\star\\, \\omega = 0\\, $ are equivalent (cf. Theorem 4.1). We also study As geometric applications, we employ the theory in [DW] and [W3], solve constant Dirichlet problems for generalized harmonic $1$-forms and $F$-harmomic maps (cf. Theorems 10.3 and 10.2), derive monotonicity formulas for $2$-balanced solutions, and vanishing theorems for $2$-moderate solutions of $\\langle\\omega, \\Delta \\omega\\rangle \\ge 0\\, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Growth estimates for generalized harmonic\\n forms on noncompact manifolds with geometric\\n applications\",\"authors\":\"S. Wei\",\"doi\":\"10.1090/conm/756/15215\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We introduce Condition W $\\\\,$(1.2) for a smooth differential form $\\\\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\\\\omega$ is a harmonic form on $M$ if and only if $\\\\omega$ is both closed and co-closed on $M\\\\, ,$ where $\\\\omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(\\\\ne 2) < 3\\\\, $ with $\\\\omega$ satisfying Condition W $\\\\,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1<q(\\\\ne 2)<3\\\\, $ satisfying Condition W $\\\\,$(1.2) is both closed and co-closed (cf. Theorem 1.1). This generalizes the work of A. Andreotti and E. Vesentini [AV] for every $L^2$ harmonic form $\\\\omega\\\\, .$ In extending $\\\\omega$ in $L^2$ to $L^q$, for $q \\\\ne 2$, Condition W $\\\\,$(1.2) has to be imposed due to counter-examples of D. Alexandru-Rugina$\\\\big($ [AR] p. 81, Remarque 3$\\\\big).$ We then study nonlinear partial differential inequalities for differential forms $ \\\\langle\\\\omega, \\\\Delta \\\\omega\\\\rangle \\\\ge 0, $ in which solutions $\\\\omega$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $\\\\omega\\\\, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $\\\\langle\\\\omega, \\\\Delta \\\\omega\\\\rangle \\\\ge 0\\\\, ,$ (ii) $\\\\Delta \\\\omega = 0\\\\, ,$ $($iii$)$$\\\\quad d\\\\, \\\\omega = d^{\\\\star}\\\\omega = 0\\\\, ,$ (iv) $\\\\langle \\\\star\\\\, \\\\omega, \\\\Delta \\\\star\\\\, \\\\omega\\\\rangle \\\\ge 0\\\\, ,$ (v) $\\\\Delta \\\\star\\\\, \\\\omega = 0\\\\, ,$ and (vi) $d\\\\, \\\\star\\\\, \\\\omega = d^{\\\\star} \\\\star\\\\, \\\\omega = 0\\\\, $ are equivalent (cf. Theorem 4.1). We also study As geometric applications, we employ the theory in [DW] and [W3], solve constant Dirichlet problems for generalized harmonic $1$-forms and $F$-harmomic maps (cf. Theorems 10.3 and 10.2), derive monotonicity formulas for $2$-balanced solutions, and vanishing theorems for $2$-moderate solutions of $\\\\langle\\\\omega, \\\\Delta \\\\omega\\\\rangle \\\\ge 0\\\\, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).\",\"PeriodicalId\":8430,\"journal\":{\"name\":\"arXiv: Differential Geometry\",\"volume\":\"21 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-10-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Differential Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/conm/756/15215\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Differential Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/conm/756/15215","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Growth estimates for generalized harmonic
forms on noncompact manifolds with geometric
applications
We introduce Condition W $\,$(1.2) for a smooth differential form $\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\omega$ is a harmonic form on $M$ if and only if $\omega$ is both closed and co-closed on $M\, ,$ where $\omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(\ne 2) < 3\, $ with $\omega$ satisfying Condition W $\,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1
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