*Published Paper*

**Inserted:** 8 nov 2016

**Last Updated:** 17 nov 2017

**Journal:** Ann. Mat. Pura Appl.

**Volume:** 193

**Number:** 4

**Pages:** 1069-1084

**Year:** 2014

**Doi:** 10.1007/s10231-012-0315-5

**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).