Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 088, 17 pages      arXiv:2307.03616      https://doi.org/10.3842/SIGMA.2023.088
Contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär for his 60th birthday

A Poincaré Formula for Differential Forms and Applications

Nicolas Ginoux ^a, Georges Habib ^ab and Simon Raulot ^c
a) Université de Lorraine, CNRS, IECL, F-57000 Metz, France
^b) Lebanese University, Faculty of Sciences II, Department of Mathematics, P.O. Box 90656 Fanar-Matn, Lebanon
^c) Université de Rouen Normandie, CNRS, Normandie Univ, LMRS UMR 6085, F-76000 Rouen, France

Received July 19, 2023, in final form October 26, 2023; Published online November 08, 2023

Abstract
We prove a new general Poincaré-type inequality for differential forms on compact Riemannian manifolds with nonempty boundary. When the boundary is isometrically immersed in Euclidean space, we derive a new inequality involving mean and scalar curvatures of the boundary only and characterize its limiting case in codimension one. A new Ros-type inequality for differential forms is also derived assuming the existence of a nonzero parallel form on the manifold.

Key words: manifolds with boundary; boundary value problems; Hodge Laplace operator; rigidity results.

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