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In this appendix, we give a short introduction to differential forms on infinite-dimensional manifolds. The main difference between the finite dimensional (or Banach) and our setting, is that it is in general impossible to interprete differential forms as (smooth) sections into certain bundles of linear forms. The reason for this is again that the topology on spaces of linear forms breaks down beyond the Banach setting. Even worse, the many equivalent ways to define differential forms in finite dimensions become inequivalent in the infinite-dimensional setting. Most notably, there is no useful way to describe differential forms as a sum of differential forms coming from a local coordinate system. We begin with the definition of a differential form. This definition is geared towards avoiding any reference to topologies on spaces of linear mappings. Then, we shall discuss differential forms on a Lie group and in particular the Maurer–Cartan form.

In this paper we study the eigenvalues and eigenfunctionsof metric measure manifolds. We prove that any eigenfunctionis $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign.We alsodiscuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.

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Appendix E - Differential Forms on Infinite-Dimensional Manifolds

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EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS

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Appendix E - Differential Forms on Infinite-Dimensional Manifolds

Summary

EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS