Fredholm, Hodge and Liouville theorems on noncompact manifolds
Fredholm, Hodge and Liouville theorems on noncompact manifolds
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非紧流形上的 Fredholm、Hodge 和 Liouville 定理
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发表时间：
1987
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通讯作者：
R. Lockhart
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作者：
R. Lockhart
在具有有限多个端点的流形上定义了一类度量，证明了与拉普拉斯算子相关的Fredholm、Liouville、Hodge和l2 -上同调定理。这些度规与渐近平移不变的度规共形。它们不一定是完整的。Fredholm的结果必然是关于加权Sobolev空间的。并证明了这些空间的嵌入定理和紧嵌入定理。在紧致黎曼流形的分析中，两个最有用的事实是：拉普拉斯流形是Fredholm，它的核由闭和共闭形式组成，这些闭和共闭形式为所有的de Rham上同调类提供了唯一的表示。人们自然想把这些结果推广到非紧化流形。在这个方向上的第一个这样的结果是由于Kodaira [9,12, p. 165]。即L2（AM, g）是dCo (Aq- 1m), dg*CO (Aq+lM), k2（AWM, g) = {aE L2(AM, g) Ida = dga* = 0）的正交直和。不幸的是，如果对流形或度规没有任何限制，人们就无法改进这一点。拉普拉斯不一定是弗雷德霍恩；l2 -谐波形式不需要闭合或共闭；即使它们是封闭和共闭的，调和L2形式的空间也不一定是有限维的（见[7]）；即使拉普拉斯算子是Fredholm， l2 -调和形式是封闭和共闭的，这些形式也不需要提供de Rham上同的唯一或全部表示。因此，非紧流形分析中的一个主要问题是M和g上的什么条件允许人们对紧流形延续Fredholm和Hodge定理，如果它们不能完全延续，它们能延续到什么程度？在Hodge定理的例子中，即L2调和形式空间的性质，这个问题最近得到了相当积极的研究（见[2- 8,11，和13]）。例如，Atiyah， Patodi和Singer在[2]中的一个结果是，如果流形具有圆柱形末端，则k 2（a M, g）自然同构于HfDR(M)中的HComp(M)的像（参见[2，命题4.9]）。在[11]中，Muller研究了紧集外为qxr +的流形上的拉普拉斯谱，由编者于1985年8月30日和1986年4月10日修订。1980年数学学科分类（1985年修订）。Primary 58GI0, 35J05。？1987美国数学学会0002-9947/87 $1.00 + $。此内容于2016年7月19日星期二04:20:24 UTC从157.55.39.45下载，所有使用须遵守http://about.jstor.org/terms
Fredholm, Liouville, Hodge, and L2-cohomology theorems are proved for Laplacians associated with a class of metrics defined on manifolds that have finitely many ends. The metrics are conformal to ones that are asymptotically translation invariant. They are not necessarily complete. The Fredholm results are, of necessity, with respect to weighted Sobolev spaces. Embedding and compact embedding theorems are also proved for these spaces. Two of the most useful facts in analysis on a compact Riemannian manifold are that the Laplacian is Fredholm and its kernel consists of closed and coclosed forms that provide unique representatives for all the de Rham cohomology classes. Naturally one would like to extend these results to noncompact manifolds. The first such result in this direction is due to Kodaira [9, 12, p. 165]. It is that L2(A M, g) is the orthogonal direct sum of dCo (Aq-1M), dg*CO (Aq+lM), and k2(AWM, g) = {aE L2(AM, g) Ida = dga* = 0). Unfortunately, if no restriction is made on the manifold or the metric, one cannot improve on this. The Laplacian need not be Fredholrn; L2-harmonic forms need not be closed or coclosed; even if they are closed and coclosed, the space of harmonic L2 forms need not be finite dimensional (see [7]); and even if the Laplacian is Fredholm and L2-harmonic forms are closed and coclosed, those forms need not provide unique or total representation of de Rham cohomology. Thus one of the main questions in analysis on noncompact manifolds is what conditions on M and g allow one to carry over the Fredholm and Hodge theorems for compact manifolds and, if they cannot be carried over completely, to what extent can they be? In the case of Hodge's theorem, i.e., properties of the space of L2 harmonic forms, this question has been quite actively investigated recently (see [2-8, 11, and 13]). For instance, one of the results of Atiyah, Patodi, and Singer in [2] is that if a manifold has cylindrical ends then k 2(A M, g) is naturally isomorphic to the image of HComp(M) in HfDR(M) (see [2, Proposition 4.9]). In [11], Muller investigates the spectrum of the Laplacian on manifolds that outside a compact set are Q x R+, with Received by the editors August 30, 1985 and, in revised form, April 10, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 58GI0, 35J05. ?1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page This content downloaded from 157.55.39.45 on Tue, 19 Jul 2016 04:20:24 UTC All use subject to http://about.jstor.org/terms