Integral Formulas for Differential Forms on Weighted Manifolds and Applications
Fida El Chami, Ola Makhoul
TL;DR
This work extends the classical Reilly formula to differential forms on weighted (Bakry–Émery) manifolds with boundary, using it to prove weighted Poincaré-type inequalities and to study boundary-value problems for forms. It develops spectral theory for these weighted manifolds, including eigenvalue estimates for the weighted boundary Laplacian and generalized buckling/clamped-plate problems, under curvature assumptions. The results yield rigidity statements and geometric bounds that connect interior Bakry–Émery curvature, boundary geometry, and the weight function, with applications to mean-curvature and immersion problems. Overall, the paper provides a cohesive framework unifying weighted differential-form analysis and boundary value problems on manifolds with density.
Abstract
In this paper, we derive a Reilly formula for differential forms on weighted manifolds with nonempty boundary. As an application of this formula, we prove a Poincaré-type inequality in the same context and explore several of its consequences. We also present weighted versions of some boundary value problems and obtain new eigenvalue estimates that extend previously known results.