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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References

1. Bourguignon J.-P., Hijazi O., Milhorat J.-L., Moroianu A., Moroianu S., A spinorial approach to Riemannian and conformal geometry, EMS Monogr. Math., European Mathematical Society (EMS), Zürich, 2015.
2. Brown J.D., York Jr. J.W., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993), 1407-1419, arXiv:gr-qc/9209012.
3. do Carmo M.P., Warner F.W., Rigidity and convexity of hypersurfaces in spheres, J. Differential Geometry 4 (1970), 133-144.
4. Duff G.F.D., Spencer D.C., Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math. 56 (1952), 128-156.
5. Friedrich T., Dirac operators in Riemannian geometry, Grad. Stud. Math., Vol. 25, American Mathematical Society, Providence, RI, 2000.
6. Gallot S., Hulin D., Lafontaine J., Riemannian geometry, 3rd ed., Universitext, Springer, Berlin, 2004.
7. Gallot S., Meyer D., Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne, J. Math. Pures Appl. 54 (1975), 259-284.
8. Ginoux N., The Dirac spectrum, Lecture Notes in Math., Vol. 1976, Springer, Berlin, 2009.
9. Hijazi O., Montiel S., A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type, Asian J. Math. 18 (2014), 489-506, arXiv:1502.04859.
10. Lawson Jr. H.B., Michelsohn M.-L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
11. Miao P., Tam L.-F., On second variation of Wang-Yau quasi-local energy, Ann. Henri Poincaré 15 (2014), 1367-1402, arXiv:1301.4656.
12. Miao P., Tam L.-F., Xie N., Critical points of Wang-Yau quasi-local energy, Ann. Henri Poincaré 12 (2011), 987-1017, arXiv:1003.5048.
13. Miao P., Wang X., Boundary effect of Ricci curvature, J. Differential Geom. 103 (2016), 59-82, arXiv:1408.2711.
14. Raulot S., Savo A., A Reilly formula and eigenvalue estimates for differential forms, J. Geom. Anal. 21 (2011), 620-640, arXiv:1003.0817.
15. Ros A., Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (1987), 447-453.
16. Savo A., On the first Hodge eigenvalue of isometric immersions, Proc. Amer. Math. Soc. 133 (2005), 587-594.
17. Schwarz G., Hodge decomposition - A method for solving boundary value problems, Lecture Notes in Math., Vol. 1607, Springer, Berlin, 1995.
18. Shi Y., Tam L.-F., Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), 79-125, arXiv:math.DG/0301047.
19. Wang M.-T., Yau S.-T., Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys. 288 (2009), 919-942, arXiv:0805.1370.