English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This article was published in the Michigan Mathematical Journal and is available here with the kind permission of the publisher.The Hodge theorem for compact manifolds states that every real cohomology class\udof a compact manifold M is represented by a unique harmonic form. That is, the\udspace of solutions to the differential equation .d Cd\ud \ud/  D 0 on L2 forms over M;\uda space that depends on the metric on M; is canonically isomorphic to the purely\udtopological real cohomology space of M: This isomorphism is enormously useful\udbecause it provides a way to transform theorems from geometry into theorems in\udtopology and vice versa. No such result holds in general for complete noncompact\udmanifolds, but in many specific cases there are Hodge-type theorems. One of the\udoldest is the description, due to Atiyah, Patodi, and Singer [1], of the space of L2\udharmonic forms on a manifold with complete cylindrical ends. By calculating the\udsolutions to the equation for harmonic forms on the cylindrical ends, they showed\udthat the space of L2 harmonic forms is isomorphic to the image of the relative cohomology\udof the manifold in the absolute cohomology. Another Hodge-type result\udwas found by Zucker [14] for a natural class of metrics called Poincaré metrics.\udThese metrics, first constructed by Cornalba and Griffiths [4], are complete Kähler\udmetrics with hyperbolic cusp-type singularities at isolated points on a Riemann\udsurface. Zucker showed that the space of L2 forms on a Riemann surface that are\udharmonic with respect one of these metrics is isomorphic to the standard cohomology\udof the surface. This result was extended by Cattani, Kaplan, and Schmid [3]\udto analogous metrics on bundles over projective varieties with singularities along\uda divisor. These metrics can be thought of as complete Kähler metrics on the noncompact\udmanifold given by removing the divisor

L2 Harmonic forms for a class of complete Kähler metrics