This paper extends the results of [GLS24], where the existence of a constant harmonic mean curvature foliation was established in the setting of a 3-dimensional asymptotically Schwarzschild manifold. Here, we generalize this construction to higher dimensions, proving the existence of foliations by constant harmonic mean curvature hypersurfaces in an asymptotically Schwarzschild manifold of arbitrary dimension. Furthermore, in 3 dimensional case, we demonstrate the local uniqueness of this foliation under a stronger decay conditions on the asymptotically Schwarzschild metric

In this paper, we prove that any two dimensional complete translating soliton in ℝ^4 with finite total curvature is a plane. It is in fact a consequence of our main result.

Under a condition that breaks the volume doubling barrier, we obtain a time polynomial structure result on the space of ancient caloric functions with polynomial growth on manifolds. As a byproduct, it is shown that the finiteness result for the space of harmonic functions with polynomial growth on manifolds in [9] and [23] are essentially sharp, except for the multi-end cases, addressing an issue raised in [11] and removing all local topological or geometric conditions on the manifold with respect to a reference point.

We prove that the tangent cone at the first blow-up time of the mean curvature flow of a closed symplectic surface in a compact Ka & uml;hler-Einstein surface consists of a finite union of planes in R4. Furthermore, when the flow develops a Type I* singularity at (X0,T), then the tangent cone is a holomorphic cone.

In this paper, we establish analytic properties of pseudoharmonic maps from pseudo-Hermitian manifolds to Riemannian manifolds. More precisely, we derive the monotonicity formula and small-energy regularity theorem for pseudoharmonic maps from 3-dimensional pseudo-Hermitian manifolds. As an application, we can prove a compactness theorem for pseudoharmonic maps with bounded energy. We also prove Liouville theorems for stable foliated harmonic maps from Heisenberg groups to spheres.

In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case.

We first demonstrate that the area preserving mean curvature flow of hypersurfaces in space forms exists for all time and converges exponentially fast to a round sphere if the integral of the traceless second fundamental form is sufficiently small. Then we show that from sufficiently large initial coordinate sphere, the area preserving mean curvature flow exists for all time and converges exponentially fast to a constant mean curvature surface in 3-dimensional asymptotically Schwarzschild spaces. This provides a new approach to the existence of foliation established by Huisken and Yau ([11]). And also a uniqueness result follows. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, the authors show that the symplectic mean curvature flow in ℂℙ2 with normal curvature pinched exists for a long time and converges to a holomorphic curve.

In this paper, we investigate the stability of the volume preserving mean curvature flow (VPMCF) and area preserving mean curvature flow (APMCF) in the Schwarzschild space. We show that if the initial hypersurface is sufficiently close to a coordinate sphere, these flows exist globally and converge smoothly to a constant mean curvature (CMC) hypersurface, namely a coordinate sphere. For asymptotically Schwarzschild space, if the initial hypersurface has pinched curvature outside of some large compact set, or more orecisely sufficiently close to an isoperimetric hypersurface, outside of some large compact set in C^2 sense, we will apply similar method combined with the center manifold analysis to see that the flow still exists for all time and converges to CMC hypersurface exponentially fast. This in particular gives an existence result for a CMC hypersurface in asymptotically flat space.

In this paper we consider a gradient flow for the L-beta-functional introduced by the authors in Han-Li-Sun (2018). We first prove that the "symplectic" property is preserved along the gradient flow. Then we prove a monotonicity formula and an epsilon-regularity theorem for the flow. As consequences, we show that the lambda-tangent cone of the flow consists of finite flat planes. Another application is that we can show that the flow exists globally and converges to a holomorphic curve if the initial surface is sufficiently close to a holomorphic curve and the ambient Kaher-Einstein surface has positive scalar curvature.

It is known that there is no a Type I singularity for the Lagrangian mean curvature flow with zero Maslov class. In this paper, we study translating solitons which are important models of Type II singularities. A necessary condition for a blow-up limit arising at a Type II singularity of a Lagrangian mean curvature flow with zero Maslov class is provided. As an application, we prove that the Lagrangian translating solitons constructed by Joyce-Lee-Tsui cannot be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class, answering an open question proposed by Joyce-Lee-Tsui and Neves-Tian , as well as we also prove that the Grim Reaper cannot be a blow-up limit for a Lagrangian mean curvature flow with zero Maslov class.

This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.

In this paper, we prove that any complete, rotationally symmetric translating soliton in C-2 cannot arise as finite time blow up limit of the symplectic mean curvature flow.

In this paper, we prove some nonexistence results for the Type II singularity to the almost calibrated Lagrangian mean curvature flow and the symplectic mean curvature flow.

In the paper, we introduce the notion of free boundary constant p-mean curvature (CPMC) surface in a 3-dimensional pseudohermitian manifold N with boundary M . It arises as the critical point of p-area among surfaces which divides N into two subsets of preassigned volumes and whose boundary is free to move in M . We introduce a stability criterion for free boundary CPMC surfaces. When M is the Pansu sphere S_1 in the Heisenberg group ℍ_1 and N is the interior of M , we find examples of free boundary CPMC surfaces which are rotationally symmetric about the t -axis of ℍ_1 . Besides the stable p-minimal disk {t=0}∩ N , there exist Pansu spherical caps which are stable free boundary CPMC surfaces intersecting S_1 . We conclude the paper by two analogous problems as those for free boundary CMC surfaces intersecting with the Euclidean 2-sphere.

In this note we will provide a gradient estimate for harmonic maps from a complete noncompact Riemannian manifold with compact boundary (which we call "Kasue manifold") into a simply connected complete Riemannian manifold with non-positive sectional curvature. As a consequence, we can obtain a Liouville theorem. We will also show the nonexistence of positive solutions to some linear elliptic equations on Kasue manifolds.

In this note, we generalize our previous results in [4] and [5] from Kähler manifold to Sasaki manifold. We prove that for a submanifold tangent to the Reeb vector field in a Sasaki manifold, the area functional, as a functional of the ambient metrics, is stationary if and only if it is an invariant submanifold. We also show that if the submanifold has dimension 3, then it is basic stable if and only if it is an invariant submanifold.

In this paper, we proved the Alexandrov-Fenchel inequalities for embedded, closed, connected and convex C2-hypersurfaces in Sn+1: Ak≥ξk,k−2(Ak−2) for any 1≤k≤n−1, where Ak is the quermassintegral (see Definition 1.1) and ξk,k−2 is the unique positive function such that the equality holds when M is a geodesic sphere.

Abstract Background: COVID-19 had caused more than 2.8 million deaths globally, and the epidemic will persist for an extended period of time. We analyzed clinical features of patients in the early stage of the epidemic, so as to deepen the understanding of the disease.Methods: In this retrospective study, we included 84 confirmed cases of COVID-19 during February 1, 2020 and March 31, 2020. Baseline data were used to classify patients as moderate (57%) or severe/critical based on Chinese protocol. We focused on analyzing the differences in chest computed tomography (CT) between the two groups. Results: Of the 84 cases, 50 were male and the median age was 69 years. 55 (65%) patients had comorbidities at admission, more in the severe/critical group (P=0.040). 94% patients had bilateral lesions on CT, up to 68% had lesions involving all lobes. Ground glass opacification (GGO) (96%), consolidation (44%), Linear opacities (50%) and Air bronchogram (23%) were the mainly lesions. The lesion was gradually absorbed over time, but imaging abnormalities can persist for a long time. Compared with moderate cases, the severe/critical group had more pulmonary consolidation changes (P=0.044) and significantly higher CT severity Score (CTSS) (P=0.040). Lymphocyte counts were significantly lower (P=0.011) and NLR were higher (P=0.029) in severe/critical cases. Conclusions: Chest CT showed bilateral and multiple GGO and consolidation mainly. After treatment, pulmonary lesions were gradually absorbed over time, and imaging abnormalities can be persistent for a long time. Lung consolidation, CTSS, comorbidity, lymphocyte counts, and NLR may be predictors of severe COVID-19.