A vanishing result for strictly p-convex domains

TL;DR

This paper re-proves a vanishing theorem for the de Rham cohomology of strictly p-convex domains in R^n, extending classical results using L^2 techniques from complex analysis and Riemannian geometry.

Contribution

It provides a new proof of a vanishing result for p-convex domains in R^n, utilizing L^2 methods originally developed for complex manifolds.

Findings

Vanishing of de Rham cohomology for strictly p-convex domains.

Extension of Andreotti-Grauert vanishing theorem to R^n.

Application of Hörmander’s L^2 techniques in Riemannian setting.

Abstract

In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Th\'eor\`eme de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in riemannian geometry, arXiv:1111.3895v1 [math.DG]). Our proof uses the L^2-techniques developed by L. H\"ormander, (An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990,) and A. Andreotti and E. Vesentini, (Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes \'Etudes Sci. Publ. Math. 25 (1965), 81--130).