Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the \mbox{Gau\ss} curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.Comment: 36page

MEAN CURVATURE FLOW OF SUBMANIFOLDS WITH SMALL TRACELESS SECOND FUNDAMENTAL FORM

Consider a family of smooth immersions F(; t) : Mn  Mn+k of submanifolds in Mn+k moving by mean curvature flow = , where  is the mean curvature vector for the evolving submanifold. We prove that for any n >-2 and k>-1, the flow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in finite time, and the corresponding normalized flow converges exponentially in the C-topology, to an n-sphere in some subspace Mn+1 of Mn+k

Northern Hemisphere mid-winter vortex-displacement and vortex-split stratospheric sudden warmings: Influence of the Madden-Julian Oscillation and Quasi-Biennial Oscillation

We investigate the connection between the equatorial Madden‐Julian Oscillation (MJO) and different types of the Northern Hemisphere mid‐winter major stratospheric sudden warmings (SSWs), i.e., vortex‐displacement and vortex‐split SSWs. The MJO‐SSW relationship for vortex‐split SSWs is stronger than that for vortex‐displacement SSWs, as a result of the stronger and more coherent eastward propagating MJOs before vortex‐split SSWs than those before vortex‐displacement SSWs. Composite analysis indicates that both the intensity and propagation features of MJO may influence the MJO‐related circulation pattern at high latitudes and the type of SSWs. A pronounced Quasi‐Biennial Oscillation (QBO) dependence is found for vortex‐displacement and vortex‐split SSWs, with vortex‐displacement (‐split) SSWs occurring preferentially in easterly (westerly) QBO phases. The lagged composites suggest that the MJO‐related anomalies in the Arctic are very likely initiated when the MJO‐related convection is active over the equatorial Indian Ocean (around the MJO phase 3). Further analysis suggests that the QBO may modulate the MJO‐related wave disturbances via its influence on the upper tropospheric subtropical jet. As a result, the MJO‐related circulation pattern in the Arctic tends to be wave number‐one/wave number‐two ~25–30 days following phase 3 (i.e., approximately phases 7–8, when the MJO‐related convection is active over the western Pacific) during easterly/westerly QBO phases, which resembles the circulation pattern associated with vortex‐displacement/vortex‐split SSWs

Clinical outcomes after vascular reconstruction using synthetic grafts for limb salvage in patients with lower extremity sarcoma: a single-center retrospective experience

IntroductionLimb-salvage surgery has become the mainstream approaches for the treatment of sarcoma in the lower extremity. In cases where the sarcoma infiltrates the primary vessel, concurrent resection of the vessels and vascular reconstruction are required to ensure sufficient resection and preservation of limb function. The objective of this study is to assess the clinical outcomes of patients who underwent vascular reconstruction utilizing synthetic grafts for limb salvage, specifically in terms of postoperative complications and limb functional status.MethodsBetween September 2016 and October 2021, 15 consecutive patients who underwent 15 arterial and 3 venous reconstruction procedures were included in this retrospective study. Incidence of postoperative morbidity, graft patency, rate of limb salvage, and overall survival of patients were analyzed.ResultsThe median follow-up was 12.5 months (range, 4.5-72.0). Graft thrombosis occurred in 5 patients (33.3%) and graft occlusion occurred in 3 patients (20.0%). The median overall survival was 28.0 months with the estimated 2-year and 5-year overall survival of 57.8% and 43.4% respectively. The 1-year and 2-year estimated patency rates of arterial reconstructions were 82.3% and 62.1%, respectively. None of the included patients with limb amputation were observed as a consequence of severe vascular complications, while two patients underwent amputation due to the repeat recurrence, resulting in a limb salvage rate of 86.7%.ConclusionOur results show that the combination of vascular reconstruction and oncologic resection is a feasible option for preserving limbs in cases of musculoskeletal sarcoma with vessel involvement in the lower extremity. When vascular reconstruction surgery is performed, synthetic substitutes can be effectively used with low perioperative morbidity and an acceptable rate of limb salvage