Mean Curvature Research Articles

Isoperimetry and the properness of weak inverse mean curvature flow

Isoperimetry and the properness of weak inverse mean curvature flow

On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini–Fogagnolo–Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum–Galloway.

Gradient conformal stationarity and the CMC condition in LRS spacetimes

Abstract We study the existence of gradient conformal Killing vectors (CKVs) in the class of locally rotationally symmetric (LRS) spacetimes which generalizes spherically symmetric spacetimes, and investigate some implications for the evolutionary character of marginally outer trapped surfaces. We first study existence of gradient CKVs via the obtention of a relationship between the Ricci curvature and the gradient of the divergence of the CKV. This provides an alternative set of equations, for which the integrability condition is obtained, to analyze the existence of gradient CKVs. A uniqueness result is obtained in the case of perfect fluids, where it is demonstrated that the Robertson-Walker solution is the unique perfect fluid solution with a nonvanishing pressure, admitting a timelike gradient CKV. The constant mean curvature condition for LRS spacetimes is also obtained, characterized by three distinct conditions which are specified by a set of three scalars. Linear combinations of these scalars, whose vanishing define the constant mean curvature condition, turn out to be related to the evolutions of null expansions of 2-spheres along their null normal directions. As such, some implications for the existence of black holes and the character of the associated horizons are obtained. It is further shown that dynamical black holes of increasing area, with a non-vanishing heat flux across the horizon, will be in equilibrium, with respect to the frame of the conformal observers.

Travel distance estimation of landslide-induced debris flows by machine learning method in Nepal Himalaya after the Gorkha earthquake

Debris flows are more likely to be triggered in the earthquake-strike areas with a widespread presence of unstable slopes, causing severe casualties and changing the surrounding natural topography. In such scenario, estimating the travel distance of debris flows becomes crucial to understand the hazardous areas. Therefore, a hybrid machine learning model (GA-XGBoost) was employed to achieve a reliable estimation of debris-flow travel distance. This model was applied to the Nepal Himalayas, the site of the 2015 Gorkha earthquake. We selected four geomorphological factors for travel distance estimation. They are the volume of failure mass (VL), the height difference between the material source center and end point of movement mass (H), the mean gradient of the travel path (J), and the mean curvature of the travel path (C). Furthermore, to eliminate the noise information and enhance stability of input data, a principal component analysis (PCA) was used to generate three principal components (PC1, PC2, and PC3) from the selected factors to serve as input variables of model development. The performance of this model was evaluated using the assessment indexes, resulting in a mean absolute percentage error (MAPE) of 8.71%, a root mean square error (RMSE) of 144.3 m, and a mean absolute error (MAE) of 86.1 m. Four empirical approaches were also introduced for comparison analysis. Our proposed model has proven to be superior and effective, as the estimated results closely match the actual values. All the results affirm the suitability of our developed model for estimating the travel distance of landslide-induced debris flows following a strong earthquake.

In Vivo Assessment of a Triple Periodic Minimal Surface Based Biomimmetic Gyroid as an Implant Material in a Rabbit Tibia Model.

Biomimetic approaches to implant construction are a rising frontier in implantology. Triple Periodic Minimal Surface (TPMS)-based additively manufactured gyroid structures offer a mean curvature of zero, rendering this structure an ideal porous architecture. Previous studies have demonstrated the ability of these structures to effectively mimic the mechanical cues required for optimal implant construction. The porous nature of gyroid materials enhances bone ingrowth, thereby improving implant stability within the body. This enhancement is attributed to the increased surface area of the gyroid structure, which is approximately 185% higher than that of a dense material of the same form factor. This larger surface area allows for enhanced cellular attachment and nutrient circulation facilitated by the porous channels. This study aims to evaluate the biological performance of a gyroid-based Ti6Al-4V implant material compared to a dense alloy counterpart. Cellular viability was assessed using the lactate dehydrogenase (LDH) assay, which demonstrated that the gyroid surface allowed marginally higher viability than dense material. The in vivo integration was studied over 6 weeks using a rabbit tibia model and characterized using X-ray, micro-CT, and histopathological examination. With a metal volume of 8.1%, the gyroid exhibited a bone volume/total volume (BV/TV) ratio of 9.6%, which is 11-fold higher than that of dense metal (0.8%). Histological assessments revealed neovascularization, in-bone growth, and the presence of a Haversian system in the gyroid structure, hinting at superior osteointegration.

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

Stability of the Generalized Lagrangian Mean Curvature Flow in Cotangent Bundle

Design and parametrization of TPMS lattice using computational and experimental approach

Triply periodic minimal surface (TPMS) based lattices are extensively explored as a scaffold design for bone regeneration. TPMS maintains zero mean curvature at each point and offers a large surface area comparable to a trabecular bone. The best four TPMS minimal surfaces (IWP. Neovius, primitive, and F-RD) were selected, designed, and fabricated using acrylonitrile butadiene styrene (ABS) resin through the stereolithography (SLA) technique. The results indicate that small changes in unit cell dimensions do not significantly alter the structure topology, which ensures stress distribution within the lattice remains relatively uniform across different unit cell sizes when the porosity level is constant. The optimal unit cell size (2 to 5 mm) and porosity (70 to 80%) significantly affect the compressive strength and surface area to volume (SA/V) ratio due to a unique arrangement of the internal architecture of each TPMS unit cell. The lattice structure (formed by stacking unit cell) of unit cell size 2.11 mm with 70% porosity exhibited a maximum compressive strength of 39.8 MPa in IWP, followed by Neovius, primitive, and F-RD-based lattice structures. Moreover, the lattice showed more stability under compression force, minimized stress concentration compared to a unit cell, and exhibited distinct deformation patterns at different strain levels during compression.

On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold

AbstractWe propose a notion of stability for constant ‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.

Kα-translators of Weingarten type in Gauss curvature flow

In this paper, we consider Kα-translating solitons in R3 and give some classifications of Kα-translating solitons fulfilling some extrinsic curvature assumptions. We prove that Kα-translating solitons with constant mean curvature are either planes or circular cylinders. We also prove that planes and circular cylinders are the only Kα-translating solitons with constant squared norm of the second fundamental form.

Curvature analysis of CI electrode arrays: a novel approach to categorize perimodiolar positions without anatomical landmarks.

Perimodiolar electrode arrays may be positioned regular, over-inserted or under-inserted into the cochlea depending on the cochlear size and shape. The study aimed to examine whether there are differences between these groups in the local curvature along the intracochlear array. Individual curvature variables were developed to categorize the groups and the relationship between the curvature and the angular insertion depth at the electrode tip was analyzed. The curvature along the intracochlear array was measured in the CBCT image of 85 perimodiolar electrodes of a single type. The mean curvature and the ratio of the mean curvature at contacts E14-16 to the mean curvature at E7-8 (bowing ratio) were calculated across the array, and its true positive rate (TPR) and false positive rate (FPR) were calculated to establish optimal threshold values to categorize the groups. 68.2% of the cases were categorized as regular positioned, 22.4% had an over-insertion and 9.4% had an under-insertion. The mean curvature was significantly weaker with under-insertion (< 342°) than with normal insertion depth (≥ 342°). With an over-insertion, the bowing ratio was < 1 and otherwise > 1. Both the mean curvature and bowing ratio were found to have an optimal threshold value with high TPR (= 1.00) and low FPR (≤ 0.06) for categorizing under-insertion and over-insertion, respectively. Curvature analysis is a useful tool to assess if a perimodiolar electrode array has been inserted deep enough into the cochlea. Independent of critical anatomical landmarks, over-inserted arrays and under-inserted arrays could be well categorized by using individual curvature variables. The results need to be validated using additional data sets.

Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials

We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in RN with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.

Local bifurcation structure and stability of the mean curvature equation in the static spacetime

We consider the curvature equation in the static spacetime, $$ \text{div} \Big(\frac{f(x)\nabla u}{\sqrt{1-f^2(x)| \nabla u|^2}}\Big) +\frac{\nabla u \nabla f(x)}{\sqrt{1-f^2(x)| \nabla u|^2}}=\lambda NH \quad\text{in }\Omega, $$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq 1\); the function \(H\) gives the mean curvature. We investigate the local bifurcation structure and stability of the solutions to this equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/48/abstr.html

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

Characterization of Trabecular Bone Microarchitecture and Mechanical Properties Using Bone Surface Curvature Distributions.

Understanding bone surface curvatures is crucial for the advancement of bone material design, as these curvatures play a significant role in the mechanical behavior and functionality of bone structures. Previous studies have demonstrated that bone surface curvature distributions could be used to characterize bone geometry and have been proposed as key parameters for biomimetic microstructure design and optimization. However, understanding of how bone surface curvature distributions correlate with bone microstructure and mechanical properties remains limited. This study hypothesized that bone surface curvature distributions could be used to predict the microstructure as well as mechanical properties of trabecular bone. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the histomorphometric parameters (e.g., BV/TV, BS, Tb.Th, DA, Conn.D, and SMI), geometric parameters (e.g., plate area PA, plate thickness PT, rod length RL, rod diameter RD, plate-to-plate nearest neighbor distance NNDPP, rod-to-rod nearest neighbor distance NNDRR, plate number PN, and rod number RN), as well as the apparent stiffness tensor of trabecular bone using various bone surface curvature distributions, including maximum principal curvature distribution, minimum principal curvature distribution, Gaussian curvature distribution, and mean curvature distribution. The results showed that the surface curvature distribution-based deep learning model achieved high fidelity in predicting the major histomorphometric parameters and geometric parameters as well as the stiffness tenor of trabecular bone, thus supporting the hypothesis of this study. The findings of this study underscore the importance of incorporating bone surface curvature analysis in the design of synthetic bone materials and implants.

Optimizing the Morphology and Solidification Behavior of Fe-Rich Phases in Eutectic Al-Si-Based Alloys with Different Fe Contents by Adding Mn Elements.

A high Fe content easily produces Fe-rich phases with a harmful morphology, resulting in a huge detrimental effect on the properties and recycling ability of Al-Si alloys. Therefore, finding ways to effectively transform Fe-rich phases to form a beneficial phase or shape is of great significance. Accordingly, Al-Si-based alloys with Fe contents ranging from 0.1 wt.% to 2.0 wt.% were modified by different Mn additions. Moreover, experiments combined with simulations were utilized to comprehensively analyze the mechanism of Mn on the morphology and microstructural evolution of Fe-rich phases from different perspectives. The current findings determine that adding different Fe contents changes the phase-transition reactions in alloys. Without Mn, and by increasing the Fe content from 0.1 wt.% to 2.0 wt.%, the Fe-rich phases gradually convert from a skeleton-shaped α-Al8Fe2Si (<0.25 wt.%) to β-Al9Fe2Si2 with a fibrous (0.5 wt.%), needle-like (1.0 wt.%) and plate-like shape without curvatures (2.0 wt.%). The maximum length and mean aspect ratio increase from 12.01 μm to 655.66 μm and from 1.96 to 84.05, while the mean curvature decreases from 8.66 × 10-2 μm-1 to 8.25 × 10-4 μm-1. The addition of 0.35 wt.% Mn promotes a new Chinese-character and petal-shaped α-Al15(FeMn)3Si2, with an atomic ratio of Fe and Mn of 1:1 when the Fe content is lower than 0.5 wt.%, while it transforms to β-Al15(FeMn)3Si2 with an atomic ratio of 5:1, presenting as a refined plate-like shape with a certain curvature, as the Fe content increases to 2.0 wt.%. Mn alters the phase reactions and increases the threshold of the Fe content required for β-Al15(FeMn)3Si2, limiting the formation and growth of them simultaneously in time and space. The enrichment of Mn atoms and solute diffusion at the growth front of β-Al15(FeMn)3Si2, as well as the strong atomic-binding ability, can deflect the growth direction of β-Al15(FeMn)3Si2 for it to have a certain curvature. Additionally, the enriched Mn atoms easily form α-Al15(FeMn)3Si2 and cause the long β-Al15(FeMn)3Si2 to be broken and refined to further reduce the damages caused to the alloy's performance. Ultimately, the maximum length and mean aspect ratio can be effectively reduced to 46.2% and 42.0%, respectively, while the mean curvature can be noticeably increased by 3.27 times with the addition of Mn.

A surgical alternative of fusiform penetrating keratoplasty for the management of severe infectious keratitis.

To describe the surgical procedure of fusiform penetrating keratoplasty (FPK) using multiple trephines of different sizes for treating patients with severe infectious keratitis. Fourteen eyes underwent FPK, and 15 eyes received conventional penetrating keratoplasty (PK) were included in the study. The best-corrected visual acuity (BCVA), refractive outcomes, endothelial cell density, and postoperative complications were recorded. The FPK group was followed for an average of 15.3±2.1mo, whereas the PK group was followed for 16.1±1.9mo. The corneal ulcers were elliptical-shaped in all 14 eyes in the FPK group. The mean BCVA (logMAR, 0.26±0.13) showed no statistically significant differences from that in the PK group (logMAR, 0.21±0.12, P>0.05) at 1y after surgery. But the mean curvature, mean astigmatism, and mean spherical equivalent in the FPK group were lower than those in the PK group (P<0.05). Peripheral anterior synechia was observed in one patient in the FPK group, whereas 6 patients in the PK group. Suture loosening and neovascularization were observed in 4 and 5 eyes in the PK group, respectively. No graft immune rejection or elevation of intraocular pressure was observed in the two groups. For patients with elliptical-shaped corneas or corneal ulcers, FPK can avoid disrupting of corneal limbus, reduce the risk of postoperative complications, and can result in satisfactory visual quality.

Hessian estimates for the Lagrangian mean curvature flow

Hessian estimates for the Lagrangian mean curvature flow

Embedding diagrams in stationary spacetimes

We find the spatial and dynamic embedding diagrams in stationary black hole spacetimes. The spatial embeddings include the NUT, pure NUT and Kerr spacetimes. In the case of pure NUT spacetime, the spatial embedding equations are solved in terms of the elliptic integrals. In other cases we obtain the spatial embedding diagrams by numerical integration of the corresponding embedding equations. These embedding diagrams are then compared through their Gaussian and mean curvatures. We also find the dynamic embedding diagrams of NUT and pure NUT spacetimes, and compare them with the dynamic embedding diagram of Schwarzschild spacetime.

Two-Level method for blind image deblurring problems

Blind image deblurring (BID) is a procedure for reducing blur and noise in a deteriorated image. In this process, the estimation of the original image, as well as the blurring kernel of the degraded image, is done without or with only partial information about the imaging system and degradation. This is an inverse problem (ill-posed) that corresponds to the direct problem of deblurring. To overcome the ill-posedness of BID and attain useful solutions, the regularization models based on mean curvature (MC) are utilized. The discretization of MC-based models often leads to a large ill-conditioned nonlinear system of equations, which is computationally expensive. Moreover, the existence of MC functionals in the governing equations of the BID model complicates the calculation of the nonlinear system. To overcome these problems, in this paper, we propose the Two-Level blind image deblurring method (TLBID). First, on the coarse-grid, we solve a small nonlinear system (with a small number of pixels) for a mesh size of H, followed by solving a large linear system of equations on the finer grid (with a large number of pixels) of size h (h≤H). On the coarse mesh, we solve the BID problem utilizing the computationally expensive MC regularization functional. After this, we interpolate the results to the finer mesh. On the finer mesh, we solve the BID problem with less computationally expensive regularization functionals such as total variation (TV) or Tikhonov. This approach produces an approximate solution of the BID equations with high accuracy, which is cost-effective. The TLBID algorithm is implemented with MATLAB, and verification and validation are carried out using benchmark problems and medical digital images.

Hybrid-feature based spherical quasi-conformal registration for AD-induced hippocampal surface morphological changes

Background and objectiveEstablishing accurate one-to-one morphological correspondence between different hippocampal surfaces is a solid foundation for the analysis of AD-induced hippocampal morphological changes. However, owing to the large variations between hippocampal surfaces, exiting registration work either fails to obtain the accurate matching of local and overall morphological features or does not preserve the bijectivity during parametric mapping. For this reason, this study proposes a hybrid-feature based spherical quasi-conformal registration (HSQR) method that can effectively maintain the diffeomorphic property while meeting the hybrid-feature matching constraints in the spherical parameter domain. MethodsThe HSQR algorithm is primarily achieved through hippocampal surface hybrid feature extraction and spherical quasi-conformal registration. First, hybrid features for a comprehensive morphological description of the hippocampal surface were established, which included essential anatomical features (landmarks) and mean curvature (intensity) features to ensure the accuracy of surface morphology alignment. Second, spherical parameterization was applied to genus-0 closed surfaces, such as the hippocampus, which maximized the preservation of the original local surface morphology through area-preserving properties. Third, a novel spherical quasi-conformal registration algorithm that can handle large deformations is established. It transforms a 3D spherical parameter domain into a 2D plane parameter domain using iterative local stereo projection to improve the efficiency of the registration algorithm. Subsequently, by controlling the Beltramin coefficient, the hybrid morphological features could be aligned while ensuring bijection before and after registration. ResultsUsing a cohort including 161 patients with amyloid-β (Aβ) positive Alzheimer disease (AD), 234 Aβ positive mild cognitive impairment (MCI) and 266 Aβ negative cognitively unimpaired (CU) individuals from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, we set up the experiment which indicated that the HSQR-based whole bilateral hippocampal atrophy features demonstrated the stronger statistical power for group morphological differences of CU vs. MCI with q-value: 0.0453 for left hippocampus and 0.0401 for right hippocampus and group morphological differences of AD vs. MCI with q-value: 0.0282 for left hippocampus and 0.0421 for right hippocampus. ConclusionsOur registration algorithm may provide a solid foundation for the accurate quantification of hippocampal surface morphological changes for the differential diagnosis and tracking of AD.