Harmonic forms on manifolds with edges

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$

Extended Hodge Theory for Fibred Cusp Manifolds

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted $L^2$ harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted $L^2$ harmonic forms are harmonic forms that are almost in the given weighted $L^2$ space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. As in that setting, in the unweighted $L^2$ case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.Comment: 26 page

Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on \RR^3 that is smooth everywhere except at a discrete set \maS of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on \RR^3 with $Z(p) > -1/4$ for p \in \maS. Also assume that $\rho$ and $Z$ are smooth outside \maS and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set \maS is finite and $V$ extends to a smooth function on the radial compactification of \RR^3 that is bounded outside a compact set containing \maS. In the periodic case, we let $\Lambda$ be the periodicity lattice and define \TT := \RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr\"odinger-type operator $H = -\Delta + V$ acting on L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55 (103), no. 2/201

A Prospective Substudy of a Randomized Controlled Trial

Malignant ascites (MA) is most commonly observed in patients scheduled for epithelial ovarian cancer (EOC) surgery and is supposed as a major risk factor promoting perioperative hemodynamic deterioration. We aimed to assess the hemodynamic consequences of MA on systemic circulation in patients undergoing cytoreductive EOC surgery. This study is a predefined post-hoc analysis of a randomized controlled pilot trial comparing intravenous solutions within a goal-directed algorithm to optimize hemodynamic therapy in patients undergoing cytoreductive EOC surgery. Ascites was used to stratify the EOC patients prior to randomization in the main study. We analyzed 2 groups according to the amount of ascites (NLAS: none or low ascites [<500 mL] vs HAS: high ascites group [>500 mL]). Differences in hemodynamic variables with respect to time were analyzed using nonparametric analysis for longitudinal data and multivariate generalized estimating equation adjusting the analysis for the randomized study groups of the main study. A total of 31 patients in the NLAS and 16 patients in the HAS group were analyzed. Although cardiac output was not different between groups suggesting a similar circulatory blood flow, the HAS group revealed higher heart rates and lower stroke volumes during surgery. There were no differences in pressure-based hemodynamic variables. In the HAS group, fluid demands, reflected by the time to reindication of a fluid challenge after preload optimization, increased steadily, whereas stroke volume could not be maintained at baseline resulting in hemodynamic instability after 1.5 h of surgery. In contrast, in the NLAS group fluid demands were stable and stroke volume could be maintained during surgery. Clinically relevant associations of the type of fluid replacement with hemodynamic consequences were particularly observed in the HAS group, in which transfusion of fresh frozen plasma (FFP) was associated to an improved circulatory flow and reduced vasopressor and fluid demands, whereas the administration of artificial infusion solutions was related to opposite effects. Malignant ascites >500 mL implies increased fluid demands and substantial alterations in circulatory blood flow during cancer surgery. Fresh frozen plasma transfusion promotes recovering hemodynamic stability in patients with malignant ascites >500 mL, in whom artificial infusion solutions could not prevent from hemodynamic deterioration

A substudy of a randomized controlled trial

Data on early markers for acute kidney injury (AKI) after noncardiovascular surgery are still limited. This study aimed to determine the diagnostic value of plasma neutrophil-gelatinase-associated lipocalin (pNGAL) and intraoperative diuresis for AKI in patients undergoing major abdominal surgery treated within a goal-directed hemodynamic algorithm. This study is a post-hoc analysis of a randomized controlled pilot trial comparing intravenous solutions within a hemodynamic goal-directed algorithm based on the esophageal Doppler in patients undergoing epithelial ovarian cancer surgery. The diagnostic value of plasma NGAL obtained at ICU admission and intraoperative diuresis was determined with respect to patients already meeting AKI criteria 6 hours after surgery (AKI6h) and to all patients meeting AKI criteria at least once during the postoperative course (AKItotal). AKI was diagnosed by the definition of the Kidney Disease Improving Global Outcome (KDIGO) group creatinine criteria and was screened up to postoperative day 3. Receiver operating characteristic curves including a gray zone approach were performed. A total of 48 patients were analyzed. None of the patients had increased creatinine levels before surgery and 14 patients (29.2%) developed AKI after surgery. Plasma NGAL was predictive for AKI6h (AUCAKI6h 0.832 (95% confidence interval [CI], 0.629–0.976), P = .001) and AKItotal (AUCAKItotal 0.710 (CI 0.511–0.878), P = .023). The gray zones of pNGAL calculated for AKI6h and AKItotal were 210 to 245 and 207 to 274 ng mL−1, respectively. The lower cutoffs of the gray zone at 207 and 210 ng mL−1 had a negative predictive value (NPV) (i.e., no AKI during the postoperative course) of 96.8% (CI 90–100) and 87.1% (CI 78–97), respectively. Intraoperative diuresis was also predictive for AKI6h (AUCAKI6h 0.742 (CI 0.581–0.871), P = .019) with a gray zone of 0.5 to 2.0 mL kg−1 h−1. At the lower cutoff of the gray zone at 0.5 mL kg−1 h−1, corresponding to the oliguric threshold, the NPV was 84.2% (78–92). This study indicates that pNGAL can be used as an early marker to rule out AKI occurring within 3 days after major abdominal surgery. Intraoperative diuresis can be used to rule out AKI occurring up to 6 hours after surgery