Fredholm, Hodge and Liouville Theorems on Noncompact Manifolds

Abstract

Fredholm, Liouville, Hodge, and e-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 1.2(/\ qM, g) is the orthogonal direct sum of dCooo(/\ q-1M), d;Co'X)(/\ q+1M), and Unfortunately, if no restriction is made on the manifold or the metric, one cannot improve on this. The Laplacian need not be Fredholm; L 2-harmonic forms need not be closed or coclosed; even if they are closed and coclosed, the space of harmonic L 2 forms need not be finite dimensional (see [7]); and even if the Laplacian is Fredholm and L 2-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 carryover 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 L 2 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 §j 2(/\ g M, g) is naturally isomorphic to the image of H~omp(M) in HZR(M) (see [2, Proposition 4.9]). In [11], Milller investigates the spectrum of the Laplacian on manifolds that outside a compact set are Q X R +, with

Deformations of asymptotically cylindricalG2-manifolds

“…[18,Theorem 7.4] is a corollary, which computes the index of self-adjoint asymptotically translation-invariant elliptic operators for small weights. This can be applied in particular to the Hodge Laplacian of an asymptotically cylindrical metric, as in [17,Section 3].…”

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

“…If M is an asymptotically cylindrical Calabi-Yau manifold, and L is an asymptotically cylindrical special Lagrangian submanifold, then the induced metric on L is itself asymptotically cylindrical. Hence, deformation results follow by combining the asymptotically cylindrical Laplace-Beltrami theory of Lockhart [27] with the McLean results. This was carried out by Salur and Todd [35].…”

Minimal hyperspheres of arbitrarily large Morse index

“…Observe that T is formally self-adjoint so we have at once that for β > 0 the operator T * has to be surjective. Now, it follows from Lockhart-McOwen theory (see [LMc85,Loc87]) that the operator T : W q+2,2…”