{'fraction_non_alphanumeric': 0.07962743128481417, 'fraction_numerical': 0.01725869783581408, 'mean_word_length': 3.548172757475083, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A vanishing result for strictly p-convex domains

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).


Introduction
A weaker condition than holomorphic convexity for domains in C n has been introduced by A. Andreotti and H. Grauert in [1], defining q-complete domains as domains in C n admitting a proper exhaustion function whose Levi form has n− p+ 1 positive eigen-values.
In a recent series of foundational papers, [6,7], and references therein, F. R. Harvey and H. B. Lawson raise the interest on generalizations of the concept of convexity for Riemannian manifolds, proving many important results for p-convex manifolds: namely, starting with a Riemannian manifold (X, g), they ask whether it admits an exhaustion function whose Hessian is positive definite or satisfies weaker positive conditions. Interpolating between the classical notions of convex functions and pluri-sub-harmonic functions, in [7] they define the class of p-pluri-sub-harmonic functions in terms of the positivity of the minors of their Hessian form, and they study pconvex domains, which can be regarded as domains in R n endowed with a smooth p-pluri-sub-harmonic proper exhaustion function. The notions of geometric pluri-sub-harmonicity and geometric convexity, introduced and studied by F. R. Harvey and H. B. Lawson in [6], is closely related to holomorphic convexity and q-completeness in the sense of A. Andreotti and H. Grauert, [1].
We are concerned in studying vanishing results for strictly p-convex domains in R n in the sense of F. R. Harvey and H. B. Lawson. More precisely, we give a proof of the following result. Theorem 3.1. Let X be a strictly p-convex domain in R n . Then H k dR (X; R) = {0} for every k ≥ p. As pointed out to us by F. R. Harvey and H. B. Lawson, the above result was already known, as a consequence of [9, Theorem 1] by J.-P. Sha, and [10, Theorem 1] by H. Wu, see also [7,Proposition 5.7]: more precisely, they prove, using Morse theory, that the existence of a smooth proper strictly p-pluri-sub-harmonic exhaustion function has consequences on the homotopy type of the domain. In spite of this, our proof differs in the techniques, which are inspired by A. Andreotti and E. Vesentini [2]: in particular, the L 2 -techniques used in our proof could be hopefully applied in a wider context, a fact which we would like to investigate further in future work.
The organization of the paper is as follows. In Section 1, we recall the main definitions introduced in [6,7], and the results proven by A. Andreotti and H. Grauert in [1]. In Section 2, we prove some useful estimates, which will be used in Section 3 to prove Theorem 3.1.
for bringing authors' attention to the results by J.-P. Sha, [9], and H. Wu, [10], and for further fruitful suggestions and remarks. They are very grateful to the anonymous referee for his/her valuable comments, which highly improved the presentation of the paper.

The notion of p-convexity by Harvey and Lawson
Following F. R. Harvey and H. B. Lawson, [7,6], firstly we recall point-wise definitions of p-positive symmetric endomorphisms, then we will turn to manifolds, and finally we will recall the notion of p-pluri-sub-harmonic (exhaustion) functions and (strictly) p-convex domains.
1.1. p-positive (sections of ) symmetric endomorphisms. Let (V, · | ·· ) be an n-dimensional real inner product for any v, w ∈ V . By means of the inner product · | ·· , the space Sym 2 (V ) is isomorphic to the space of the · | ··symmetric endomorphisms of V : given A ∈ Sym 2 (V ), we denote by Finally, given a · | ·· -symmetric endomorphism E ∈ Hom (V, V ), let sgn (E) denote the number of non-negative eigenvalues of E.
Notice that, given A ∈ Sym 2 (V ), and given two inner products on V inducing respectively the isomorphisms G 1 and G 2 , then there holds sgn G −1 where λ 1 , . . . , λ n are the eigenvalues of G −1 A. 7,6]).
• Let V be a R-vector space endowed with an inner product · | ·· . Denote the space of p-positive forms of k th branch on V as • Let (X, g) be a Riemannian manifold. Define the space of p-positive sections of k th branch of the bundle Sym 2 (T X) of symmetric endomorphisms of T X as 1.2. p-pluri-sub-harmonic functions. In order to introduce an exhaustion of a given Riemannian manifold, we focus on special p-positive symmetric 2-forms, those arising from the Hessian of smooth functions.
Thus, let (X, g) be a Riemannian manifold and let u be a smooth real valued function on X. Let ∇ denote the Levi-Civita connection of the Riemannian metric g, and let where V and W are smooth sections of the tangent bundle T X. Thus, Hess u(x) ∈ Sym 2 (T x X), for any x ∈ X.
• The space PSH (k) p (X, g) := u ∈ C ∞ (X; R) : Hess u ∈ P (k) p (X, g) , is called the space of p-pluri-sub-harmonic functions of k th branch on X.
• The space int PSH (k) p (X, g) := u ∈ C ∞ (X; R) : Hess u ∈ int P (k) p (X, g) , p (X, g) denotes the interior of P (k) p (X, g)) is called the space of strictly p-pluri-sub-harmonic functions of k th branch on X.

(Strictly) p-convexity.
We are now ready to recall the concept of p-convexity, which is central in [7].
Let (X, g) be a Riemannian manifold. Let K ⊆ X be a compact set. The p-convex hull of K is given by ). Let (X, g) be a Riemannian manifold. Then, X is called p-convex if, for any compact set K ⊆ X, then K PSH (1) p (X, g) is relatively compact in X.
Define the p-core of X, [6, Definition 4.1], as Definition 1.4 ([6]). Let (X, g) be a Riemannian manifold. Then, X is called strictly p-convex if (i) Core p (X, g) = ∅ and, (ii) for any compact set K ⊆ X, then K PSH (1) p (X, g) is relatively compact in X.
1.5. The p-convexity and the q-completeness. All along the definitions of the previous section, the special case that we had in mind is the following classical construction in Complex Analysis.
In [1], A. Andreotti and H. Grauert pointed out the following concept.
A. Andreotti and H. Grauert, in [1], studied domains of C n admitting strictly q-pluri-sub-harmonic exhaustion functions (the so-called q-complete domains), proving a vanishing theorem for the higher-degree Dolbeault cohomology groups of such domains; then A. Andreotti and E. Vesentini, in [2], re-prove the same result extending the L 2 -techniques by L. Hörmander, [8].
Thus, in the same vein as A. Andreotti and H. Grauert, we would consider domains X in R n endowed with an exhaustion function u ∈ C ∞ (X; R) whose Hessian is in int P (1) p (X, g) , proving a vanishing result for the higher-degree de Rham cohomology groups for strictly p-convex domains in the sense of F. R. Harvey and H. B. Lawson.

Vanishing of the de Rham cohomology for strictly p-convex domains
Let X be an oriented Riemannian manifold of dimension n, and denote by g its Riemannian metric and by vol its volume. The Riemannian metric g induces, for every x ∈ X, a point-wise scalar product · |·· gx : and, for k ∈ N, define L 2 φ X; ∧ k T * X as the completion of the space C ∞ c X; ∧ k T * X of smooth k-forms with compact support, with respect to the metric induced by · L 2 For any k ∈ N, let L 2 loc X; ∧ k T * X denote the space of k-forms f whose restriction f ⌊ K to every compact set K ⊆ X belongs to L 2 K; ∧ k T * X .
For every φ 1 , φ 2 ∈ C 0 (X; R), the operator is densely-defined and closed; denote by its adjoint, which is a densely-defined closed operator.
Recall that, on a domain X in R n , fixed k ∈ N, s ∈ N, and φ ∈ C ∞ (X; R), the Sobolev space W s,2 . . , ℓ n ) ∈ N n such that ℓ 1 + · · · + ℓ n ≤ s, and for every strictly increasing multi-index I such that |I| = k. The space W s,2 loc X; ∧ k T * X is defined as the space of k-forms f whose restriction f ⌊ K to every compact set K ⊆ X belongs to W s,2 K; ∧ k T * X . As a matter of notation, the symbol |I|=k denotes the sum over the strictly increasing multi-indices I :=: (i 1 , . . . , i k ) ∈ N k (that is, the multi-indices such that 0 < i 1 < · · · < i k ) of length k. Given I 1 and I 2 two multi-indices of length k, let sign I 1 I 2 be the sign of the permutation I 1 I 2 if I 1 is a permutation of I 2 , and zero otherwise.
2.1. Some preliminary computations. Let X be a domain in R n , that is, an open connected subset of R n , endowed with the metric and the volume induced, respectively, by the Euclidean metric and the standard volume of R n .
The following lemma gives an explicit expression of the adjoint d * φ2, φ1 : Let v :=: Proof. By definition of d * φ2, φ1 , for every u ∈ dom d, one has d u | v L 2 . Hence, consider u :=: The statement follows by computing For any fixed φ ∈ C ∞ (X; R) and for any j ∈ {1, . . . , n}, define the operator The following lemma states that δ φ j is the adjoint of ∂ ∂x j in L 2 φ X; ∧ 0 T * X , and computes the commutator between δ φ j and ∂ ∂x k (compare with, e.g., [8, pages 83-84]).
For every J such that |J| = k − 1, the previous equality gives By the arithmetic mean-geometric mean inequality, one gets where C :=: C(k, n) ∈ N depends on k and n only. Now, using Lemma 2.2, one computes sign Hence, in view of (3), (2), (1), we get concluding the proof.
Remark 2.4. The argument in the proof of Proposition 2.3 actually proves the following stronger estimate, which will be used in the regularization process in Theorem 3.1. Let X be a domain in R n and φ, ψ ∈ C ∞ (X; R). Consider Then, for any η :=: where C :=: C(k, n) ∈ N is a constant depending just on k and n.

Proof of the main theorem
We are ready to prove the following vanishing theorem for the higher-degree de Rham cohomology groups of a strictly p-convex domain in R n (for a different proof, involving Morse theory, compare [9, Theorem 1] by J.-P. Sha, and [10, Theorem 1] by H. Wu, see also [7,Proposition 5.7]).
Proof. We are going to prove that every d-closed k-form η ∈ C ∞ X; ∧ k T * X is d-exact, namely, there exists α ∈ C ∞ X; ∧ k−1 T * X such that η = d α; the statement of the theorem is a direct consequence of this result. Let us split the proof in the following steps.