Mean Curvature Research Articles

Statistical Submanifolds Equipped with F-Statistical Connections

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with F-statistical connections are examined. The equations of Gauss and Codazzi of F-statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to F-statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared F-mean curvature of a statistical submanifold of statistical manifolds.

A note on parallel mean curvature surfaces and Codazzi operators

A note on parallel mean curvature surfaces and Codazzi operators

Some Gradient Estimates for Nonlinear Heat-Type Equations on Smooth Metric Measure Spaces with Compact Boundary

In this paper we prove some Hamilton type and Li–Yau type gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure space with compact boundary. The geometry of the space in terms of lower bounds on the weighted Bakry–Émery Ricci curvature tensor and weighted mean curvature of the boundary are key in proving generalized local and global gradient estimates. Various applications of these gradient estimates in terms of parabolic Harnack inequalities and Liouville type results are discussed. Further consequences to some specific models informed by the nature of the nonlinearities are highlighted.

A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation

A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation

Homothetic $$\alpha $$-Self-Similar Solutions to the Mean Curvature Flow in Minkowski 3-Space

Homothetic $$\alpha $$-Self-Similar Solutions to the Mean Curvature Flow in Minkowski 3-Space

On minimal surfaces in ℍ2 × ℝ space

Abstract A surface is minimal if the mean curvature ℋ mean {\mathcal{H}_{\rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ 2 × ℝ {\mathbb{H}^{2}\times\mathbb{R}} . In particular, we completely classify minimal surfaces.

Radial solutions for Neumann problems involving prescribed mean curvature operator in a ball and in an annular domain

Using topological transversality method together with barrier strip technique and cut-off technique, we obtain new existence and uniqueness results of radial solutions to the Neumann problems involving prescribed mean curvature operator div∇v1+|∇v|2=f|x|,v,dvdrinΩ,∂v∂n=0on∂Ω,where Ω={x∈RN:R1<|x|<R2}(0≤R1<R2,N≥2), f:[R1,R2]×R2→R is continuous. Meanwhile, we demonstrate the importance of our results through an illustrative example.

The class of grim reapers in [formula omitted]

We study translators of the mean curvature flow in the product space H2×R. In H2×R there are three types of translations: vertical translations due to the factor R and parabolic and hyperbolic translations from H2. A grim reaper in H2×R is a translator invariant by a one-parameter group of translations. The variety of translators and translations in H2×R makes the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of H2×R with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.

An Attention-Based Hemispheric Relation Inference Network for Perinatal Brain Age Prediction.

Brain anatomical age is an effective feature to assess the status of the brain, such as atypical development and aging. Although some deep learning models have been developed for estimating infant brain age, the performance of these models was unsatisfactory because few of them considered the developmental characteristics of brain anatomy during the perinatal period-the most rapid and complex developmental stage across the lifespan. The present study proposed an attention-based hemispheric relation inference network (HRINet) that takes advantage of the nature of brain structural lateralization during early development. This model captures the inter-hemispheric relationship using a graph attention mechanism and transmits lateralization information as features to describe the interactive development between bilateral hemispheres. The HRINet was used to estimate the brain age of 531 preterm and full-term neonates from the Developing Human Connectome Project (dHCP) database based on two metrics (mean curvature and sulcal depth) characterizing the folding morphology of the cortex. Our results showed that the HRINet outperformed other benchmark models in fitting the perinatal brain age, with mean absolute error of 0.53 and determination coefficient of 0.89. We also verified the generalizability of the HRINet on an extra independent dataset collected from the Gansu Provincial Maternity and Child-care Hospital. Furthermore, by applying the best-performing model to an independent dataset consisting of 47 scans of preterm infants at term-equivalent age, we showed that the predicted age was significantly lower than the chronological age, suggesting a delayed development of premature brains. Our results demonstrate the effectiveness and generalizability of the HRINet in estimating infant brain age, providing promising clinical applications for assessing neonatal brain maturity.

Calculation of creep deformation of stiffness-degraded RC beams after strengthening with increased cross-section

The strengthening of existing reinforced concrete beams (RCBs) has mainly focused on short-term mechanical properties such as flexural resistance and shear resistance, while long-term deflection of RCBs has seldom been studied. In this study, six RCBs with different types and degrees of damage were designed and strengthened by the section enlargement method. Mid-span deflection and concrete strain were used to evaluate the influence of stiffness degradation due to fatigue or corrosion damage on the creep behavior of strengthened RCBs (SRCBs). The experiment results revealed that time-varying curves of creep deformation from fatigue or from corrosion-damaged SRCBs were consistent among the six specimens. Creep in a damaged SRCB was found to increase faster than that in an undamaged beam, and to reach over 90 % of final creep in about six months. Based on the mean curvature method and creep models, taking fatigue and corrosion damage as examples of stiffness degradation, a calculation method for creep deformation of SRCBs was established, and sensitivity analysis of stress levels, reinforcement height, consolidation time, and other parameters were carried out. The research results can provide a theoretical basis and guidance for evaluating creep deformation of stiffness-degraded RCBs after reinforcement with increased cross-section.

Mesoporous Metal Sponges Produced by Explosive Decomposition.

The formation of mesoporous gold sponges by explosive decomposition of 'knallgold' (also known as 'fulminating' gold) is studied. Proof-of-principle experiments are conducted and then the phenomena are further investigated using 'toy physics' molecular dynamics simulations. The simulations invoked various ratios of a volatile Lennard-Jones element G and a noble metal element N. In both experiment and simulation the morphology of the resulting sponge is found to depend on the stoichiometry of the starting material. As the mole fraction of G (χG) is increased from 0.5 to close to 1.0 in the simulations, the morphology of the sponges changes from closed to open, with a corresponding increase in the average mean curvature from 0 to +0.12 inverse Lennard-Jones length (L) units. The average Gaussian curvature of the simulated sponges is always negative, with the minimum value of 0.05 L-2 being found for χG≈0.65. In broad agreement with experiment, sponge formation in the simulations is bounded by stoichiometry; no sponges form if χG is <0.52, for χG between 0.52 and 0.70 the sponge is characterized by vermicular cavities whereas classic bicontinuous fibrous sponges form for 0.70<χG<0.85 and, finally, discrete particles result if χG>0.85.

Mean Exit Times from Submanifolds with Bounded Mean Curvature

Mean Exit Times from Submanifolds with Bounded Mean Curvature

Securing IoT network with hybrid evolutionary lion intrusion detection system: a composite motion optimisation algorithm for feature selection and ensemble classification

ABSTRACT Networks connected to the Internet of Things (IoT) are often vulnerable to attacks. Several existing methods in the intrusion detection system for securing IoT have been presented with ensemble classifier, but it does not accurately classify attack, and also it takes high computation time. With intention of solving the security issues, Intrusion Detection System using Hybrid Evolutionary Lion and Balancing Composite Motion Optimisation Algorithm espoused feature selection with Ensemble Classifier (IDS-IoT-Hybrid ELOA-BCMOA-Ensemble-DT-LSVM-RF-XGBoost) is proposed for Securing IoT Network. At first, data were accumulated from the NSL-KDD data set. Afterward, data is fed to pre-processing, where it restored missing value using mean curvature flow method. At feature selection, optimum features are compiled under Hybrid Evolutionary Lion and Balancing Composite Motion Optimisation Algorithm. Based upon the optimum features, intruders of IoT data are categorised as denial-of-service (DoS), probe, remote to local attack (R2L), user to root attack (U2R), normal (no attack) with the help of Ensemble classifier. Proposed IDS-IoT-Hybrid ELOA-BCMOA-Ensemble-DT-LSVM-RF-XGBoost approach is constructed utilising Python. Then, proposed IDS-IoT-Hybrid ELOA-BCMOA-Ensemble-DT-LSVM-RF-XGBoost approach attains 21.11%, 19.58%, 24.61% and 9.52% higher accuracy; 94.47%, 93.95%, 93.08% and 90.59% lower error rate, and 62.94%, 36.69%, 64.17% and 50.97% less computation time analysed with existing models.

A De Lellis–Müller type estimate on the Minkowski lightcone

We prove an analogue statement to an estimate by De Lellis–Müller in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document}-close to a round surface provided the trace-free part of a scalar second fundamental form A is sufficiently small in L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}. To determine the correct intrinsically round cross section of reference, we define an associated 4-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis–Topping.

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Our investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, we delve into the condition that determines when equality is achieved within the inequality. Furthermore, we explore a number of implications of our findings.

A note on new weighted geometric inequalities for hypersurfaces in ℝⁿ

In this note, we prove a family of sharp weighed inequalities which involve weighted k k -th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in R n \mathbb {R}^n . This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao’s results.

On the Shape of Hypersurfaces with Boundary Which Have Zero Fractional Mean Curvature

We consider compact hypersurfaces with boundary in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} that are the critical points of the fractional area introduced by Paroni et al. (Commun Pure Appl Anal 17:709–727, 2018). In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} of dimension N-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-2$$\\end{document} and lies in a hyperplane H⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\subset {\\mathbb {R}}^N$$\\end{document}. Then we show that the critical points coincide with a smooth manifold N⊂H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}\\subset H$$\\end{document} of dimension N-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-1$$\\end{document} with ∂N=Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial {\\mathcal {N}}= \\Gamma $$\\end{document}. Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document} and Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2$$\\end{document} of dimension N-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-2$$\\end{document} and suppose that Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document} lies in a hyperplane H perpendicular to the xN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_N$$\\end{document}-axis and that Γ2=Γ1+deN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2 = \\Gamma _1 + d \\, e_N$$\\end{document} (d>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d >0$$\\end{document} and eN=(0,⋯,0,1)∈RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e_N = (0,\\cdots ,0,1) \\in {\\mathbb {R}}^N$$\\end{document}). Then, assuming that Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document} has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds N1⊂H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}_1 \\subset H$$\\end{document} and N2⊂H+deN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}_2 \\subset H + d \\, e_N$$\\end{document} of dimension N-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N-1$$\\end{document} with ∂Ni=Γi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial {\\mathcal {N}}_i = \\Gamma _i$$\\end{document} for i∈{1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i \\in \\{1,2\\}$$\\end{document}. Moreover, the interior of the critical points does not intersect the boundary of the convex hull in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} of Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document} and Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2$$\\end{document}, while this can occur in the codimension-one situation considered by Dipierro et al. (Proc Am Math Soc 150:2223–2237, 2022). We also obtain a quantitative bound which may tell us how different the critical points are from N1∪N2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}_1 \\cup {\\mathcal {N}}_2$$\\end{document}. Finally, in the same setting as in the second case, we show that, if d is sufficiently large, then the critical points are disconnected and, if d is sufficiently small, then Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document} and Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2$$\\end{document} are in the same connected component of the critical points when N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N \\ge 3$$\\end{document}. Moreover, by computing the fractional mean curvature of a cone whose boundary is Γ1∪Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1 \\cup \\Gamma _2$$\\end{document}, we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.

Finite Total Curvature and Soap Bubbles With Almost Constant Higher-Order Mean Curvature

Abstract Given $ n \geq 2 $ and $ k \in \{2, \ldots , n\} $, we study the asymptotic behaviour of sequences of bounded $C^{2}$-domains, whose $ k $-th mean curvature functions converge in $ L^{1} $-norm to a constant. Under certain curvature assumptions, we prove that finite unions of mutually tangent balls are the only possible limits with respect to convergence in volume and perimeter. The key novelty of our statement lies in the fact that we do not assume bounds on the exterior or interior touching balls.

Examples of cosmological spacetimes without CMC Cauchy surfaces

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

Visualization of Isometric Deformations of Helicoidal CMC Surfaces

Helicoidal surfaces of constant mean curvature were fully described by do Carmo and Dajczer. However, the obtained parameterizations are given in terms of somewhat complicated integrals, and as a consequence, not many examples of such surfaces are visualized. In this paper, by using these methods in some particular cases, we provide several interesting visualizations involving these surfaces, mostly as an isometric deformation of a rotational surface. We also give interpretations of some older results involving helicoidal surfaces, motivated by the work carried out by Malkowsky and Veličković. All of the graphics in this paper were created in Wolfram Mathematica.