Positive Mean Curvature Research Articles

Obstacle Problems with Double Boundary Condition for Least Gradient Functions in Metric Measure Spaces

AbstractIn the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain $$\Omega $$ Ω of functions bounded between two obstacle functions inside $$\Omega $$ Ω , and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $$\Omega $$ Ω is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of $$ \varepsilon $$ ε -weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.

Flame self-interactions in MILD combustion of homogeneous and inhomogeneous mixtures

Flame self-interaction (FSI) events in Moderate or Intense Low-Oxygen Dilution (MILD) combustion of homogeneous and inhomogeneous mixtures of methane and oxidiser have been analysed using three-dimensional Direct Numerical Simulations (DNS). The simulations have been conducted at the same global equivalence ratio (⟨ϕ⟩=0.8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle = 0.8$$\\end{document}) for different levels of O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {O_2}$$\\end{document} concentration (dilution) and initial turbulence intensities. It has been reported that both homogeneous and inhomogeneous mixture MILD combustion cases exhibit significant occurrences of FSI events, with the peak frequency of FSI events occurring towards the burned gas side in all cases. Moreover, the frequency of FSI events increases with increasing dilution level and turbulence intensity, but the presence of mixture inhomogeneity leads to a reduction in total FSI events. In all cases, the cylindrical FSI topologies (i.e. tunnel formation and tunnel closure) were found to have a higher likelihood of occurrence compared to spherical FSI topologies (i.e. unburned and burned gas pockets). The geometries of FSI topologies were also analysed using the mean and Gaussian curvatures. It has been shown that the inward propagating spherical FSI topologies (i.e. unburned gas pockets) are associated with negative mean curvature, while outward propagating spherical FSI topologies (i.e. burned gas pockets) are associated with positive mean curvature. Moreover, tunnel formation (tunnel closure) FSI topologies predominantly exhibit either elliptic geometries with positive (negative) mean curvature or hyperbolic saddle geometries with negative (positive) mean curvature. It has been shown for the first time in MILD combustion that the mean values of kinematic restoration and dissipation terms in the transport equation of the magnitude of the reaction progress variable conditional upon the reaction progress variable tend to cancel each other in the vicinity of the critical points associated with cylindrical topologies. Thus, the singular contributions in these terms, which are obtained from analytical descriptions in the vicinity of tunnel formation and tunnel closure topologies, do not affect the balance equation of the magnitude of the gradient of the reaction progress variable. Consequently, there is no need for a separate model treatment for singularities in modelling approaches based on the magnitude of the gradient of the reaction progress variable. The FSI events in the reaction dominated and propagating flame regions of MILD combustion have also been analysed for the first time. It has been found that more FSI events occur in the reaction dominated region, particularly towards the burned gas side. However, the majority of spherical FSI topologies are found in the propagating flame region. The findings from this study indicate that turbulence intensity, dilution level and mixture inhomogeneity effects need to be considered in any attempt to extend flame surface-based modelling approaches to MILD combustion.

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Membrane-Active Peptides Attack Cell Membranes in a Lipid-Regulated Curvature-Generating Mode.

Membrane-active peptides (MAPs) exhibit great potential in biomedical applications due to their unique ability to overcome the cell membrane barrier. However, the interactions between MAPs and membranes are complex, and little is known about the possibility of MAP action being specific to certain types of membranes. In this study, a combination of molecular dynamics simulations and theoretical analysis was utilized to investigate the interactions between typical MAPs and realistic cell membrane systems. Remarkably, the simulations revealed that MAPs can attack membranes by generating and sensing positive mean curvature, which is dependent on lipid composition. Furthermore, theoretical calculations demonstrated that this lipid-regulated curvature-based membrane attack mechanism is an integrated result of multiple effects, including peptide-induced membrane wedge and softening effects, the lipid shape effect, the area-difference elastic effect, and the boundary edge effect of formed peptide-lipid nanodomains. This study enhances our comprehension of MAP-membrane interactions and highlights the potential for developing membrane-specific MAP-based agents.

ІНФІНІТЕЗИМАЛЬНІ ДЕФОРМАЦІЇ ПОВЕРХОНЬ ІЗ ЗАДАНОЮ ЗМІНОЮ ТЕНЗОРА РІЧЧІ

In three-dimensional Euclidean space, we study the problem of the existence of an infinitesimal first-order deformation of single-connected regular surfaces with a predetermined change in the Ricci tensor. It is shown that for surfaces of nonzero Gaussian curvature, this problem is reduced to the study and solution of a system of seven equations (including differential equations) with respect to seven unknown functions, each solution of which determines a vector field that is a univariate function (with an accuracy of a constant vector) and can be interpreted as a moment-free stress state of equilibrium of a loaded shell. For regular surfaces of non-zero Gaussian and mean curvatures, the problem is reduced to finding solutions to one second-order partial differential equation with respect to two unknown functions. Given one of these functions, the resulting equation will in general be a nonhomogeneous second-order partial differential equation (nonhomogeneous Weingarten differential equation). It is proved that any regular surface of positive Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor in a sufficiently small region. In this case, the tensor fields will be represented by an arbitrary and predefined regular function. By considering the Neumann problem, it is shown that a single-connected regular surface of elliptic type of positive Gaussian and negative mean curvature with a regular boundary under a certain boundary condition admits, in general, an infinitesimal first-order deformation with a predetermined change in the Ricci tensor. In this case, the tensor fields will be determined uniquely. For surfaces of negative Gaussian and non-zero mean curvature, the resulting inhomogeneous partial differential equation with second-order partial differentials will be of hyperbolic type with known coefficients and right-hand side. The Darboux problem is considered for this equation. It is proved that any regular surface of negative Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor. Tensor fields are expressed through a given function of two variables and through two arbitrary regular functions of one variable. Keywords: infinitesimal deformation, Ricci tensor, tensor fields, Gaussian curvature, mean curvature.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.

Vapor mixing in turbulent vaporizing flows

Scalar mixing and evaporation processes are analyzed in a compressible two-phase homogeneous and isotropic turbulence configuration. To this end, the fully compressible Navier–Stokes equations are solved using a pressure-based method and a mass-conservative interface capturing method (CLSVOF). Temperature and vapor mass fraction transport equations are coupled with the momentum equation via the evaporation rate: i.e. heat and mass transfer has an effect on the flow dynamic through the generation of a Stefan flow at the interface. In this formalism, a pressure equation is developed to include the acoustic, thermal dilatation and compressibility effects. Moreover, the method can consider several gas structures in the same domain, each with independent temperature, density, and thermodynamic pressure. The latter is important to simulate atomization, where interaction among the liquid structures (such as breakup or coalescence) can generate multiple gas inclusions. This work is one of the first to describe with high fidelity the vaporization and turbulent mixing occurring in dense two-phase flows.A statistical analysis of the vapor mass fraction field is performed. The obtained results reveal the influence of the non-homogeneous scalar source at the interface, which depend directly on the local interface temperature, on the PDF shape. Under these conditions, the vapor mass fraction PDF deviates from the Gaussian and beta PDF shapes, frequently used for turbulent combustion modeling. Furthermore, access to the local surface averaged vapor mass fraction and evaporation rate at the interface allows a statistical analysis of these variables. Results show a large dispersion of both quantities, especially at the early times of the simulations, which contradict the usual assumption of a constant vapor mass fraction (or vaporization rate) at the interface used in the literature in standard vaporization models. Finally, the influence of the mean interface curvature on the evaporation rate is quantified and discussed. Large evaporation rate are present when the interface is convex (positive mean curvature). On the contrary, local saturation zones are present near concave interface (negative mean curvature), reducing the evaporation rate magnitude.

Magainin 2 and PGLa in bacterial membrane mimics IV: Membrane curvature and partitioning

We previously reported that the synergistically enhanced antimicrobial activity of magainin 2 (MG2a) and PGLa is related to membrane adhesion and fusion. Here, we demonstrate that equimolar mixtures of MG2a and L18W-PGLa induce positive monolayer curvature stress and sense, at the same time, positive mean and Gaussian bilayer curvatures already at low amounts of bound peptide. The combination of both abilities—membrane curvature sensing and inducing—is most likely the base for the synergistically enhanced peptide activity. In addition, our coarse-grained simulations suggest that fusion stalks are promoted by decreasing the free-energy barrier for their formation rather than by stabilizing their shape. We also interrogated peptide partitioning as a function of lipid and peptide concentration using tryptophan fluorescence spectroscopy and peptide-induced leakage of dyes from lipid vesicles. In agreement with a previous report, we find increased membrane partitioning of L18W-PGLa in the presence of MG2a. However, this effect does not prevail to lipid concentrations higher than 1 mM, above which all peptides associate with the lipid bilayers. This implies that synergistic effects of MG2a and L18W-PGLa in previously reported experiments with lipid concentrations >1 mM are due to peptide-induced membrane remodeling and not their specific membrane partitioning.

Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces

We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight (2019) that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that are approximable from above and below by continuous functions. We also show that for each f\in L^1(\partial\Omega) , there is a least gradient function in \Omega whose trace agrees with f at all points of continuity of f , and so we obtain existence of solutions for boundary data which are continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan (2014), who constructed an L^1 -function on the unit circle which has no least gradient solution in the unit disk in \mathbb{R}^2 . Modifying the example of Spradlin and Tamasan, we show that the space of solvable L^1 -functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition.

Computational Design of Self‐Actuated Surfaces by Printing Plastic Ribbons on Stretched Fabric

AbstractWe introduce a new mechanism for self‐actuating deployable structures, based on printing a dense pattern of closely‐spaced plastic ribbons on sheets of pre‐stretched elastic fabric. We leverage two shape‐changing effects that occur when such an assembly is printed and allowed to relax: first, the incompressible plastic ribbons frustrate the contraction of the fabric back to its rest state, forcing residual strain in the fabric and creating intrinsic curvature. Second, the differential compression at the interface between the plastic and fabric layers yields a bilayer effect in the direction of the ribbons, making each ribbon buckle into an arc at equilibrium state and creating extrinsic curvature. We describe an inverse design tool to fabricate low‐cost, lightweight prototypes of freeform surfaces using the controllable directional distortion and curvature offered by this mechanism. The core of our method is a parameterization algorithm that bounds surface distortions along and across principal curvature directions, along with a pattern synthesis algorithm that covers a surface with ribbons to match the target distortions and curvature given by the aforementioned parameterization. We demonstrate the flexibility and accuracy of our method by fabricating and measuring a variety of surfaces, including nearly‐developable surfaces as well as surfaces with positive and negative mean curvature, which we achieve thanks to a simple hardware setup that allows printing on both sides of the fabric.

Opening nodes in the DPW method: Co-planar case

We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$

We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, being the mean curvature larger than $\frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $\mathbb H^2\times\mathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $\mathbb{S}^2\times\mathbb{R}$, which have constant mean curvature $H>\frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $H\leq\frac{1}{2}$ at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times\mathbb{R}$.

End-exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape.

Theories of strongly stretched polymer brushes, particularly the parabolic brush theory, are valuable for providing analytically tractable predictions for the thermodynamic behavior of surface-grafted polymers in a wide range of settings. However, the parabolic brush limit fails to describe polymers grafted to convex curved substrates, such as the surfaces of spherical nanoparticles or the interfaces of strongly segregated block copolymers. It has previously been shown that strongly stretched curved brushes require a boundary layer devoid of free chain ends, requiring modifications of the theoretical analysis. While this "end-exclusion zone" has been successfully incorporated into the descriptions of brushes grafted onto the outer surfaces of cylinders and spheres, the behavior of brushes on surfaces of arbitrary curvature has not yet been studied. We present a formulation of the strong-stretching theory for molten brushes on the surfaces of arbitrary curvature and identify four distinct regimes of interest for which brushes are predicted to possess end-exclusion zones, notably including regimes of positive mean curvature but negative Gaussian curvature. Through numerical solutions of the strong-stretching brush equations, we report predicted scaling of the size of the end-exclusion zone, the chain end distribution, the chain polarization, and the free energy of stretching with mean and Gaussian surface curvatures. Through these results, we present a comprehensive picture of how the brush geometry influences the end-exclusion zones and exact strong-stretching free energies, which can be applied, for example, to model the full spectrum of brush geometries encountered in block copolymer melt assembly.

On the number of critical points of stable solutions in bounded strip-like domains

In this paper we show that there exists a family of domains Ωε⊆RN with N≥2, such that the stable solution of the problem{−Δu=g(u)in Ωεu>0in Ωεu=0on ∂Ωε admits k critical points with k≥2. Moreover the sets Ωε are star-shaped and “close” to a strip as ε→0. Next, if g(u)≡1 and N≥3 we exhibit a family of domains Ωε with positive mean curvature and solutions uε which have k critical points with k≥2. In this case, the domains Ωε turn out to be “close” to a cylinder as ε→0.

Area bound for surfaces in generic gravitational field

Abstract We define an attractive gravity probe surface (AGPS) as a compact 2-surface $S_\alpha$ with positive mean curvature $k$ satisfying $r^a D_a k / k^2 \ge \alpha$ (for a constant $\alpha>-1/2$) in the local inverse mean curvature flow, where $r^a D_a k$ is the derivative of $k$ in the outward unit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy the areal inequality $A_\alpha \le 4\pi [ ( 3+4\alpha)/(1+2\alpha) ]^2(Gm)^2$, where $A_{\alpha}$ is the area of $S_\alpha$ and $m$ is the Arnowitt–Deser–Misner mass. Equality is realized when the space is isometric to the $t=$ constant hypersurface of the Schwarzschild spacetime and $S_\alpha$ is an $r=\mathrm{constant}$ surface with $r^a D_a k / k^2 = \alpha$. We adapt the two methods of the inverse mean curvature flow and the conformal flow. Therefore, our result is applicable to the case where $S_\alpha$ has multiple components. For anti-de Sitter (AdS) spaces, a similar inequality is derived, but the proof is performed only by using the inverse mean curvature flow. We also discuss the cases with asymptotically locally AdS spaces.

FIB-SEM as a Volume Electron Microscopy Approach to Study Cellular Architectures in SARS-CoV-2 and Other Viral Infections: A Practical Primer for a Virologist.

The visualization of cellular ultrastructure over a wide range of volumes is becoming possible by increasingly powerful techniques grouped under the rubric “volume electron microscopy” or volume EM (vEM). Focused ion beam scanning electron microscopy (FIB-SEM) occupies a “Goldilocks zone” in vEM: iterative and automated cycles of milling and imaging allow the interrogation of microns-thick specimens in 3-D at resolutions of tens of nanometers or less. This bestows on FIB-SEM the unique ability to aid the accurate and precise study of architectures of virus-cell interactions. Here we give the virologist or cell biologist a primer on FIB-SEM imaging in the context of vEM and discuss practical aspects of a room temperature FIB-SEM experiment. In an in vitro study of SARS-CoV-2 infection, we show that accurate quantitation of viral densities and surface curvatures enabled by FIB-SEM imaging reveals SARS-CoV-2 viruses preferentially located at areas of plasma membrane that have positive mean curvatures.

Anisotropic and crystalline mean curvature flow of mean-convex sets

We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.

The role of differential diffusion during early flame kernel development under engine conditions – part II: Effect of flame structure and geometry

From experimental spark ignition (SI) engine studies, it is known that the slow-down of early flame kernel development caused by the (Le > 1)-property of common transportation-fuel/air mixtures tends to increase cycle-to-cycle variations (CCV). To improve the fundamental understanding of the complex phenomena inside the flame structure of developing flame kernels, an engine-relevant DNS database is investigated in this work. Conclusive analyses are enabled by considering equivalent flame kernels and turbulent planar flames computed with Le > 1 and Le=1. In Part I of the present study (Falkenstein et al., Combust. Flame, 2020), a reduced representation of the local mixture state was proposed for the purpose of this analysis. Fluctuations in heat release rate attributed to differential diffusion were found to be governed by the parameters local enthalpy, local equivalence ratio, and H-radical mass fraction. Here, a coupling relation for the diffusion-controlled mixture parameter local enthalpy with local flame geometry and structure is derived, characterized by the key parameters κ and |∇c|/|∇c|lam. The analysis shows that the large positive global mean curvature intrinsic to the flame kernel configuration may detrimentally affect the local mixture state inside the reaction zone, particularly during the initial flame kernel development phase. External energy supply by spark ignition may effectively bridge over this critical stage, which causes the impact of global mean flame kernel curvature to be small under the present conditions compared to the overall effect of Le ≠ 1 observed in a statistically planar flame. Once ignition effects have decayed, the mixture state inside the reaction zone locally exhibits an identical dependence on |∇c| as in a strained laminar flame. This implies that differential diffusion effects at Karlovitz numbers representative for part-load conditions are not weakened by small-scale turbulent mixing, which is undesirable for the engine application, but can be favorable in terms of modeling.

Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation

We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of “lower energy” solutions to the above equation fails when the dimension of the manifold is not less than 62.

Human ESCRT-III polymers assemble on positively curved membranes and induce helical membrane tube formation

Endosomal sorting complexes for transport-III (ESCRT-III) assemble in vivo onto membranes with negative Gaussian curvature. How membrane shape influences ESCRT-III polymerization and how ESCRT-III shapes membranes is yet unclear. Human core ESCRT-III proteins, CHMP4B, CHMP2A, CHMP2B and CHMP3 are used to address this issue in vitro by combining membrane nanotube pulling experiments, cryo-electron tomography and AFM. We show that CHMP4B filaments preferentially bind to flat membranes or to tubes with positive mean curvature. Both CHMP2B and CHMP2A/CHMP3 assemble on positively curved membrane tubes. Combinations of CHMP4B/CHMP2B and CHMP4B/CHMP2A/CHMP3 are recruited to the neck of pulled membrane tubes and reshape vesicles into helical “corkscrew-like” membrane tubes. Sub-tomogram averaging reveals that the ESCRT-III filaments assemble parallel and locally perpendicular to the tube axis, highlighting the mechanical stresses imposed by ESCRT-III. Our results underline the versatile membrane remodeling activity of ESCRT-III that may be a general feature required for cellular membrane remodeling processes.