Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne-P\'{o}lya-Weinberger and Yang universal inequalities.Comment: To appear in Calculus of variations and Partial Differential Equation

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on compact embedded strictly pseudoconvex CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented contact forms $\mathcal{P}_+$. We show that the functionals are continuous with respect to a natural topology on $\mathcal{P}_+$. Using a simple adaptation of the standard Kato-Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical contact forms, in a generalized sense, for the functionals. We give a necessary (also sufficient in some situations) condition for a contact form to be critical. Finally, we present explicit examples of critical contact form on both homogeneous and non-homogeneous CR manifolds.Comment: 19 pages. Comments are welcom

On the continuity of the eigenvalues of a sublaplacian

International audienceWe study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology

Gemcitabine and treatment of diffuse large B-cell lymphoma in relapsed or refractory elderly patients: A prospective randomized trial in Algeria

Context: Support for non-Hodgkin\u2032s lymphoma (NHL) with large cells that is refractory or relapsed after first-line chemotherapy poses a greater therapeutic problem with bone marrow transplant therapy or when old age is a contra-indication for high-dose chemotherapy, especially among developing countries such as Algeria. Aim: To show that the regimen, including gemcitabine, could be more effective in treating elderly patients with diffuse large B-cell lymphoma (DLBCL) in relapse / refractory, without complete remission, when compared with the ESHAP (etoposide, cisplatine, solumedrol, aracytine) regimen. Materials and Methods: Ninety-six patients in the age group of 60-70 years were volunteers for a prospective randomized single-blind study, carried out for three years. Patients were divided into two groups by the drawing of lots. The first group (GA, n = 48, relapse; n = 27 [56.3%], refractory; n = 21 [43.7%]) received treatment with ESHAP protocol and the second one (GB, n = 48, relapse; n = 28 [58%], refractory; n = 20 [42%]) with GPD (gemcitabine, dexamethasone, cisplatine) protocol. Results: The overall response rates and mean survival at three years were significantly higher among patients subjected to GPD treatment compared with those subjected to ESHAP treatment (63% vs. 55%, P = 0.01 and 20.5% [95% CI 16.5-24.5] vs. 11.8% [8.9-14.6], respectively). Additionally, three-year progression-free and event-free survival rates were 20.5% (16.3-24) and 19.7% (15.9-23.5), respectively, for the GPD regimen and 10.9% (8.2-13.7) and 11.1% (95% CI 8.5-13.7), respectively, for the ESHAP regimen. Moreover, the GPD regimen was associated with improving overall survival (RR=2.02, 95% CI 1.59-2.56; P = 0.000), event-free survival (2.03, 1.64-2.52; P < 0.001) and progression-free survival (1.86, 1.46-2.37; P < 0.001). Conclusion: In cases of contra-indication for high-dose chemotherapy for elderly patients with DLBCL, without complete remission, the Gemcitabine-based therapy protocol represents a more effective and less toxic than that of ESHAP

Effects of hospital facilities on patient outcomes after cancer surgery: an international, prospective, observational study

Background Early death after cancer surgery is higher in low-income and middle-income countries (LMICs) compared with in high-income countries, yet the impact of facility characteristics on early postoperative outcomes is unknown. The aim of this study was to examine the association between hospital infrastructure, resource availability, and processes on early outcomes after cancer surgery worldwide.Methods A multimethods analysis was performed as part of the GlobalSurg 3 study-a multicentre, international, prospective cohort study of patients who had surgery for breast, colorectal, or gastric cancer. The primary outcomes were 30-day mortality and 30-day major complication rates. Potentially beneficial hospital facilities were identified by variable selection to select those associated with 30-day mortality. Adjusted outcomes were determined using generalised estimating equations to account for patient characteristics and country-income group, with population stratification by hospital.Findings Between April 1, 2018, and April 23, 2019, facility-level data were collected for 9685 patients across 238 hospitals in 66 countries (91 hospitals in 20 high-income countries; 57 hospitals in 19 upper-middle-income countries; and 90 hospitals in 27 low-income to lower-middle-income countries). The availability of five hospital facilities was inversely associated with mortality: ultrasound, CT scanner, critical care unit, opioid analgesia, and oncologist. After adjustment for case-mix and country income group, hospitals with three or fewer of these facilities (62 hospitals, 1294 patients) had higher mortality compared with those with four or five (adjusted odds ratio [OR] 3.85 [95% CI 2.58-5.75]; p<0.0001), with excess mortality predominantly explained by a limited capacity to rescue following the development of major complications (63.0% vs 82.7%; OR 0.35 [0.23-0.53]; p<0.0001). Across LMICs, improvements in hospital facilities would prevent one to three deaths for every 100 patients undergoing surgery for cancer.Interpretation Hospitals with higher levels of infrastructure and resources have better outcomes after cancer surgery, independent of country income. Without urgent strengthening of hospital infrastructure and resources, the reductions in cancer-associated mortality associated with improved access will not be realised

Le spectre du sous-laplacien sur les variétés CR strictement pseudoconvexes

The purpose of this thesis is to study the spectrum of sublaplacians on compact strictly pseudoconvex CR manifolds. We prove the discreteness of the Dirichlet spectrum of the sublaplacian $\Delta_b$ on a smoothly bounded domain $\Omega \subset M$ in a strictly pseudoconvex CR manifold M satisfying Poincaré inequality. We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology. We establish inequalities for the eigenvalues of $\Delta_b$ on compact strictly pseudoconvex CR manifolds (possibly with nonempty boundary) %$C^2$ semi-isometric maps into a Euclidean space or a Heisenberg group. Our estimates extend those obtained by P-C. Niu \& H. Zhang \cite{NiZh} for the Dirichlet eigenvalues of the sublaplacian on a bounded domain in the Heisenberg group, in the spirit of Payne-P\'{o}lya -Weinberger and Yang inequalities. We establish a new lower bound on the first nonzero eigenvalue $\lambda_1 (\theta )$ of the sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$ carrying a contact form $\theta$ whose Tanaka-Webster connection has Ricci curvature bounded from below.Le but de cette thèse est d'étudier le spectre du sous-laplacien sur les variétés CR strictement peusdoconvexes. Nous prouvons que le spectre du sous-laplacien $\Delta_b$ est discret sur un domaine borné $\Omega \subset M$ d'une variété CR strictement pseudoconvexe qui satisfait l'inégalité de Poincaré, sous les conditions de Dirichlet au bord. Nous étudions le comportement des valeurs propres du sous-laplacien $\Delta_b$ sur une variété CR strictement pseudoconvexe compacte $M$, en tant que fonctionnelle sur l'espace ${\mathcal P}_+$ de formes de contact positivement orientées sur $M$ en dotant ${\mathcal P}_+$ d'une topologie métrique naturelle. Nous établissons des inégalités pour les valeurs propres de $\Delta_b$ sur des variétés CR strictement pseudoconvexes ( éventuellement à bord non vide). Nos estimations prolongent les résultats obtenus par P-C. Niu \& H. Zhang \cite{NiZh} pour les valeurs propres du sous-laplacien avec conditions de Dirichlet au bord sur un domaine borné du groupe de Heisenberg, et sont dans l'esprit des inégalités de Payne-P\'{o}lya-Weinberger et Yang. Nous obtenons une nouvelle borne inférieure sur la première valeur propre non nulle $\lambda_1 (\theta )$ du sous-laplacien $\Delta_b$ sur une variété CR strictement pseudoconvexe compacte $M$ munie d'une forme de contact $\theta$ dont la connexion de Tanaka-Webster est à courbure de Ricci minorée

Spectrum of sublaplacians on strictly pseudoconvex CR manifolds

Le but de cette thèse est d’étudier le spectre du sous-laplacien sur les variétés CR strictement pseudoconvexe. Nous prouvons que le spectre du sous-laplacien ∆_b est discret sur un domaine borné Ω⊂M d’une variété CR strictement pseudoconvexe qui satisfait l’inégalité de Poincaré, sous les conditions de Dirichlet au bord. Nous étudions le comportement des valeurs propres du sous-laplacien ∆_b sur une variété C] strictement pseudoconvexe compacte M, en tait que fonctionnelle sur l’espace P_+ de formes de contact positivement orientées sur M en dotant P_+ d’une topologie métrique naturelle. Nous établissons des inégalités pour les valeurs propres de ∆_b sur des variétés CR strictement pseudoconvexes (éventuellement à bord non vide). Nos estimations prolongent les résultats d, tenus par P-C. Niu \& H. Zhang \cite{NiZh) pour les valeurs propres du sous-laplacien avec conditions de Dirichlet au bord sur un domaine borné du groupe de Heisenberg, et sont dans l’esprit des inégalités de Payne-PV(o)lya-Weinberger et Yang. Nous obtenons une nouvelle borne inférieure sur la première valeur propre non nulle λ1_(θ) du sous-laplacien ∆_b sur une variété CR strictement pseudoconvexe compacte M munie d’une forme de contact θ dont la connexion de Tanaka-Webster est à courbure de Ricci minorée.The purpose of this thesis is to study the spectrum of sublaplacians on compact strictly pseudoconvex CR manifolds. We prove the discreteness of the Dirichiet spectrum of the sublaplacian ∆_b on a smoothly bounded domain Ω⊂M in a strictly pseudoconvex CR manifold M satisfying Poincaré inequality. We study the behavior of the eigenvalues of a sublaplacian ∆_b on a compact strictly pseudoconvex CR manifol as functions on the set P_+ of positively oriented contact forms on M by endowing P_+ with a natural metric topology. We establish inequalities for the eigenvalues of ∆_b on compact strictly pseudoconvex CR manifolds (possibly with nonempty boundary) %C^2 semi-isometric maps into a Euclidean space or a Heisenberg group. Our estimates extend those obtained by P-C. Niu \& H. Zhang \cite{NiZh} for the Dirichlet eigenvalues 0f the sublaplacian on a bounded domain in the Heisenberg group, in the spirit of Payne-P\’{o)lya -Weinberger and Yang inequalities. We establish a new lower bound on the first nonzero eigen value λ1_(θ) of the sublaplacian ∆_b on a compact strictly pseudoconvex CR manifold dollarMdollar carrying a contact form θ whose Tanaka-Webster connection has Ricci curvature bounded from below

Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold: Eigenvalues of the sub-Laplacian

International audienceGiven a compact strictly pseudoconvex CR manifold $M$, we study the differentiability of the eigenvalues of the sub-Laplacian $\Delta_{b,\theta}$ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) $\theta$ on $M$, under conformal deformations of $\theta$. As a first application, we show that the property of having only simple eigenvalues is generic with respect to $\theta$, i.e. the set of structures $\theta$ such that all the eigenvalues of $\Delta_{b,\theta}$ are simple, is residual (and hence dense) in the set of all compatible positively oriented contact forms on $M$. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian structure of the functional $\theta\mapsto \lambda_k(\theta)$, where $\lambda_k(\theta)$ is the $k$-th eigenvalue of the sub-Laplacian $\Delta_{b,\theta}$, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical

Dirichlet and Neumann eigenvalue problems on CR manifolds

International audienc

THE FIRST POSITIVE EIGENVALUE OF THE SUB-LAPLACIAN ON CR SPHERES

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure θ on the CR sphere S 2n+1 ⊂ C n+1 , achieves its maximum when θ is the standard contact form