Mean Curvature Research Articles

Measurement of outer diameter parameters of manual grinding cable joints based on 3D point cloud processing

For the problems associated with low measurement accuracy caused by surface shrinkage, uneven grinding, and irregular shapes in manual grinding cable joints, as well as the limitations of existing machine vision measurement techniques for the outer diameter parameters, this paper proposes a non-contact outer diameter parameters measurement method of manual grinding cable joints based on three-dimensional (3D) point cloud processing. Firstly, statistical filtering is used to remove discrete noise, and coordinate transformation preprocessing is conducted to ordered the point cloud. Secondly, a bidirectional point cloud slicing method is proposed based on the cable joint structure and point cloud distribution to preserve the original features for effective measurement. Subsequently, the normalized mean curvature is utilized as the weight of point cloud coordinate averaging to construct key points for surface slices, which reduces the amount of data in the sliced point cloud and improves the matching efficiency. Finally, the key points are matched according to the position of the surface slices, and the distance of point pairs is calculated to complete the measurement of the outer diameter parameters. Experiments and error analysis are conducted on multiple manual grinding 110 kV and 500 kV real cable joints. The absolute error in the measurement of the outer diameter is less than 0.2 mm, with a relative error within 1 %. The absolute error of the XY-axis outer diameter deviation is less than 0.1 mm. The experiments demonstrate that the proposed method exhibits superior performance in cable joint outer diameter measurement.

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $$\mathbb {R}^2$$

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $$\mathbb {R}^2$$

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

Global Existence of a Mean Curvature Flow in a Cone

Global Existence of a Mean Curvature Flow in a Cone

Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three

abstract: We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in $\mathbb{R}^{3}$. The method is motivated by the integral method of Warren and Yuan. The new observation here is that we construct a ``Lagrangian'' graph which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.

Classification of complete gradient conformal mean curvature solitons

A Riemannian manifold (M,∂M,g,f) is called a gradient conformal mean curvature soliton if a smooth function f satisfies(0.1){(λ−Hg)g=∇2fand∂f∂νg=0,on ∂M;Rg=0,inM, where ∇2f is the Hessian of f with respect to the metric induced by g on ∂M and νg is the outward unit normal vector with respect to metric g. In this study, we classified nontrivial complete gradient conformal mean curvature solitons.

Mean curvature flow solitons in warped products: nonexistence, rigidity and stability

Mean curvature flow solitons in warped products: nonexistence, rigidity and stability

Rotationally invariant translators of the mean curvature flow in Einstein's static universe

In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.

The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows

The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows

Multimodal imaging and electrophysiological study in the differential diagnosis of rest tremor

IntroductionDistinguishing tremor-dominant Parkinson's disease (tPD) from essential tremor with rest tremor (rET) can be challenging and often requires dopamine imaging. This study aimed to differentiate between these two diseases through a machine learning (ML) approach based on rest tremor (RT) electrophysiological features and structural MRI data.MethodsWe enrolled 72 patients including 40 tPD patients and 32 rET patients, and 45 control subjects (HC). RT electrophysiological features (frequency, amplitude, and phase) were calculated using surface electromyography (sEMG). Several MRI morphometric variables (cortical thickness, surface area, cortical/subcortical volumes, roughness, and mean curvature) were extracted using Freesurfer. ML models based on a tree-based classification algorithm termed XGBoost using MRI and/or electrophysiological data were tested in distinguishing tPD from rET patients.ResultsBoth structural MRI and sEMG data showed acceptable performance in distinguishing the two patient groups. Models based on electrophysiological data performed slightly better than those based on MRI data only (mean AUC: 0.92 and 0.87, respectively; p = 0.0071). The top-performing model used a combination of sEMG features (amplitude and phase) and MRI data (cortical volumes, surface area, and mean curvature), reaching AUC: 0.97 ± 0.03 and outperforming models using separately either MRI (p = 0.0001) or EMG data (p = 0.0231). In the best model, the most important feature was the RT phase.ConclusionMachine learning models combining electrophysiological and MRI data showed great potential in distinguishing between tPD and rET patients and may serve as biomarkers to support clinicians in the differential diagnosis of rest tremor syndromes in the absence of expensive and invasive diagnostic procedures such as dopamine imaging.

Time-global existence of generalized BV flow via the Allen–Cahn equation

We show that a mean curvature flow obtained as the limit of the Allen–Cahn equation is not only a Brakke flow but also a generalized BV flow proposed by Stuvard and Tonegawa (2024).

Non-blind constraint image deblurring problem with mean curvature functional

Non-blind constraint image deblurring problem with mean curvature functional

On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector

In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2.

Curvature-mediated rapid extravasation and penetration of nanoparticles against interstitial fluid pressure for improved drug delivery

Elevated interstitial fluid pressure (IFP) within pathological tissues (e.g., tumors, obstructed kidneys, and cirrhotic livers) creates a significant hindrance to the transport of nanomedicine, ultimately impairing the therapeutic efficiency. Among these tissues, solid tumors present the most challenging scenario. While several strategies through reducing tumor IFP have been devised to enhance nanoparticle delivery, few approaches focus on modulating the intrinsic properties of nanoparticles to effectively counteract IFP during extravasation and penetration, which are precisely the stages obstructed by elevated IFP. Herein, we propose an innovative solution by engineering nanoparticles with a fusiform shape of high curvature, enabling efficient surmounting of IFP barriers during extravasation and penetration within tumor tissues. Through experimental and theoretical analyses, we demonstrate that the elongated nanoparticles with the highest mean curvature outperform spherical and rod-shaped counterparts against elevated IFP, leading to superior intratumoral accumulation and antitumor efficacy. Super-resolution microscopy and molecular dynamics simulations uncover the underlying mechanisms in which the high curvature contributes to diminished drag force in surmounting high-pressure differentials during extravasation. Simultaneously, the facilitated rotational movement augments the hopping frequency during penetration. This study effectively addresses the limitations posed by high-pressure impediments, uncovers the mutual interactions between the physical properties of NPs and their environment, and presents a promising avenue for advancing cancer treatment through nanomedicine.

On complete embedded translating solitons of the mean curvature flow that are of finite genus

On complete embedded translating solitons of the mean curvature flow that are of finite genus

Global intracranial arterial tortuosity is associated with intracranial atherosclerotic burden

The effect of arterial tortuosity on intracranial atherosclerosis (ICAS) is not well understood. This study aimed to evaluate the effect of global intracranial arterial tortuosity on intracranial atherosclerotic burden in patients with ischemic stroke. We included patients with acute ischemic stroke who underwent magnetic resonance angiography (MRA) and classified them into three groups according to the ICAS burden. Global tortuosity index (GTI) was defined as the standardized mean curvature of the entire intracranial arteries, measured by in-house vessel analysis software. Of the 516 patients included, 274 patients had no ICAS, 140 patients had a low ICAS burden, and 102 patients had a high ICAS burden. GTI increased with higher ICAS burden. After adjustment for age, sex, vascular risk factors, and standardized mean arterial area, GTI was independently associated with ICAS burden (adjusted odds ratio [adjusted OR] 1.33; 95% confidence interval [CI] 1.09–1.62). The degree of association increased when the arterial tortuosity was analyzed limited to the basal arteries (adjusted OR 1.48; 95% CI 1.22–1.81). We demonstrated that GTI is associated with ICAS burden in patients with ischemic stroke, suggesting a role for global arterial tortuosity in ICAS.

Diffuse-interface approximation and weak–strong uniqueness of anisotropic mean curvature flow

Abstract The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Volume preserving mean curvature flow of L2-almost umbilical hypersurfaces in hyperbolic space

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an [Formula: see text]-almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)].

Mean curvature flow with generic low-entropy initial data

Mean curvature flow with generic low-entropy initial data

Classifications of THA-surfaces in I^3

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. A surface M2 in I3 is a THA-surface of first type if it can be parameterized by r(s, t) = (s, t, Af(s + at)g(t) + B(f(s + at) + g(t))). A surface M2 in I3 is a THA- surface of second type if it can be parameterized by r(s, t) = (s, Af(s + at)g(t) + B(f(s + at) + g(t)), t), where A and B are non-zero real numbers [16, 17, 18]. In this paper, we classify two types THA-surfaces in the 3-dimensional isotropic space I3 and study THA-surfaces with zero curvature in I3.