Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Patient outcomes after laminotomy, hemilaminectomy, laminectomy and laminectomy with instrumented fusion for spinal canal stenosis: a propensity score-based study from the Spine Tango registry.

PURPOSE To compare patient outcomes and complication rates after different decompression techniques or instrumented fusion (IF) in lumbar spinal stenosis (LSS). METHODS The multicentre study was based on Spine Tango data. Inclusion criteria were LSS with a posterior decompression and pre- and postoperative COMI assessment between 3 and 24 months. 1,176 cases were assigned to four groups: (1) laminotomy (n = 642), (2) hemilaminectomy (n = 196), (3) laminectomy (n = 230) and (4) laminectomy combined with an IF (n = 108). Clinical outcomes were achievement of minimum relevant change in COMI back and leg pain and COMI score (2.2 points), surgical and general complications, measures taken due to complications, and reintervention on the index level based on patient information. The inverse propensity score weighting method was used for adjustment. RESULTS Laminotomy, hemilaminectomy and laminectomy were significantly less beneficial than laminectomy in combination with IF regarding leg pain (ORs with 95% CI 0.52, 0.34-0.81; 0.25, 0.15-0.41; 0.44, 0.27-0.72, respectively) and COMI score improvement (ORs with 95% CI 0.51, 0.33-0.81; 0.30, 0.18-0.51; 0.48, 0.29-0.79, respectively). However, the sole decompressions caused significantly fewer surgical (ORs with 95% CI 0.42, 0.26-0.69; 0.33, 0.17-0.63; 0.39, 0.21-0.71, respectively) and general complications (ORs with 95% CI 0.11, 0.04-0.29; 0.03, 0.003-0.41; 0.25, 0.09-0.71, respectively) than laminectomy in combination with IF. Accordingly, the likelihood of required measures was also significantly lower after laminotomy (OR 0.28, 95% CI 0.17-0.46), hemilaminectomy (OR 0.28, 95% CI 0.15-0.53) and after laminectomy (OR 0.39, 95% CI 0.22-0.68) in comparison with laminectomy with IF. The likelihood of a reintervention was not significantly different between the treatment groups. DISCUSSION As already demonstrated in the literature, decompression in patients with LSS is a very effective treatment. Despite better patient outcomes after laminectomy in combination with IF, caution is advised due to higher rates of surgical and general complications and consequent required measures. Based on the current study, laminotomy or laminectomy, rather than hemilaminectomy, is recommendable for minimum relevant pain relief.