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1.
$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature 下载免费PDF全文
Jiuru Zhou 《数学研究》2021,54(4):396-406
In general, the space of $L^p$ harmonic forms $\mathcal{H}^k(L^p(M))$ and reduced $L^p$ cohomology $H^k(L^p(M))$ might be not isomorphic on a complete Riemannian manifold $M$, except for $p=2$. Nevertheless, one can consider whether $\mathrm{dim}\mathcal{H}^k(L^p(M))<+\infty$ are equivalent to $\mathrm{dim}H^k(L^p(M))<+\infty$. In order to study such kind of problems, this paper obtains that dimension of space of $L^p$ harmonic forms on a hypersurface in unit sphere with finite total curvature is finite, which is also a generalization of the previous work by Zhu. The next step will be the investigation of dimension of the reduced $L^p$ cohomology on such hypersurfaces. 相似文献
2.
Martin Werner Licht 《Foundations of Computational Mathematics》2017,17(4):1085-1122
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincaré–Friedrichs-type inequalities will be studied in a subsequent contribution. 相似文献
3.
Derchyi Wu 《Annals of Global Analysis and Geometry》1997,15(2):133-139
We prove an existing theorem of homotopic harmonic diffeomorphisms between complete surfaces of finite total curvature and nontrivial genus. This is a generalization of a theorem of Jost and Schoen [4]. 相似文献
4.
Michael Roitman 《Journal of Pure and Applied Algebra》2004,194(3):329-345
In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A0 of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A0-linear. We also show that in this case A is simple if and only if A0 is a field. 相似文献
5.
We introduce a natural generalization of the notion of harmonic functions on a Cayley tree and use some properties of the group representation of the Cayley tree to describe the set of harmonic functions periodic with respect to normal subgroups of finite index. 相似文献
6.
Harmonic functions are studied on complete Riemannian manifolds. A decay estimate is given for bounded harmonic functions of variable sign. For unbounded harmonic functions of variable sign, relations are derived between growth properties and nodal domains. On Riemannian manifolds of nonnegative Ricci curvature, it has been conjectured that harmonic functions, having at most a given order of polynomial growth, must form a finite dimensional vector space. This conjecture is established in certain special cases. 相似文献
7.
8.
Julia Hartmann Anne V. Shepler 《Transactions of the American Mathematical Society》2008,360(1):123-133
Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for groups with a polynomial ring of invariants over arbitrary fields using a ramification formula of Benson and Crawley-Boevey.
9.
There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S
1-valued function defined on the boundary of a bounded regular domain of R
n
. When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence,
a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic
map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S
1-valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic
extension.
Received December 3, 1998 / final version received May 10, 1999 相似文献
10.
A harmonic function defined in a cone and vanishing on the boundary is expanded into an infinite sum of certain fundamental harmonic functions. The growth conditions under which it is reduced to a finite sum of them are discussed. 相似文献
11.
Allan J Fryant 《Journal of Mathematical Analysis and Applications》1978,66(3):599-605
Using the Bergman B3 integral operator, the growth of harmonic functions of three variables which have no finite singularities is considered. Growth of the harmonic function is related to the growth of its B3 associate, and an expression for the order and bounds on the type of an entire harmonic function are obtained in terms of its coefficients in a spherical harmonic expansion. 相似文献
12.
In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disc. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi-type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions. 相似文献
13.
14.
A uniqueness theorem for two distinct non-constant meromorphic functions that share three values of finite weights is proved, which generalizes two previous results by H.X. Yi, and X.M. Li and H.X. Yi. As applications of it, many known results by H.X. Yi and P. Li, etc. could be improved. Furthermore, with the concept of finite-weight sharing, extensions on Osgood-Yang's conjecture and Mues' conjecture, and a generalization of some prevenient results by M. Ozawa and H. Ueda, ect. could be obtained. 相似文献
15.
Amy F. Szczepański 《代数通讯》2013,41(11):4159-4169
A result of Artin, Small, and Zhang is used to show that a Noetherian algebra over a commutative, Noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, Noetherian associated graded ring. This result is extended to show that if an algebra over a commutative Noetherian ring has a locally finite, Noetherian associated graded ring, then the intersection of the powers of the Jacobson radical is nilpotent. The proofs rely on a weak generalization of generic flatness and some observations about G-rings. 相似文献
16.
Kenji Koike 《Journal of Number Theory》2007,124(1):123-141
In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM. 相似文献
17.
Xin ZHANG 《Frontiers of Mathematics in China》2018,13(5):1189-1214
We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem. 相似文献
18.
Rodrigo P. Gomez 《manuscripta mathematica》1997,93(1):325-335
In this article we study the behavior of harmonic maps from compact connected Lie groups with bi-invariant metrics into a
Hilbert orthogonal group. In particular, we will demonstrate that any such harmonic map always has image contained within
someO(n),n<∞. Since homomorphisms are a special subset of the harmonic maps we get as a corollary an extension of the Peter-Weyl theorem,
namely, that every representation of a connected compact Lie group is finite dimensional.
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. 相似文献
19.
Louis H. Blake 《Probability Theory and Related Fields》1988,79(3):405-430
Summary This paper introduces and studies a generalization of the notion of martingale which allows for a generalization of the concept of a Markov chain and a generalization of the concept of harmonic and superharmonic functions. The theory is supported by examples and techniques that suggest the natural character of the material developed.Deceased. Please address correspondence on Prof. Magda Peligrad; Department of Mathematical Sciences, University of Cincinnati, Mail Location 25, Cincinnati, OH 45221 USA 相似文献
20.
In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian
manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative
Ricci curvature is conformally equivalent to either a plane R
2 or a cylinder R × S
1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].
The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L
2-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative
sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to
Miyaoka [20] and Palmer [21].
Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science
and Technology, Japan.
The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea.
Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga
840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea 相似文献