Integral Of Mean Curvature Research Articles

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 Crofton’s type formulas and the solid angle of convex sets

AbstractHere we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.

Geometric integral formulas of cylinders within slabs

We present new expressions for the integrals of mean curvature of domains in Rn by means of sections with cylinders. Then, we combine these expressions with the corresponding version of the invariant density of affine subspaces in Rn, in order to obtain pseudo-rotational formulae for all the integrals of mean curvature of ∂K. As particular cases, we present pseudo-rotational integral formulas for the volume, area, integral of mean curvature, and Euler-Poincaré characteristic of a connected domain of R3, whose boundary is a surface, considering slabs in R3 whose central plane passes through a fixed point, and cylinders contained in these slabs.

Geometric inequalities involving three quantities in warped product manifolds

In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family concerns the k-th boundary momentum, area and weighted volume, and has applications to Weinstock-type inequalities for Steklov or Wentzell eigenvalues on star-shaped mean convex domains. This generalizes the main results of [12] and confirms a conjecture posed therein. The second family involves a weighted k-th mean curvature integral and two distinct quermassintegrals. It generalizes the authors' recent work [33] with G. Wheeler and V.-M. Wheeler, and the second author's recent work [43] with T. Zhou, in two different aspects.

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.

Total mean curvatures of Riemannian hypersurfaces

Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.

3D grain growth in nanocrystalline Al via molecular dynamics: Influence of size, topology and integral mean curvature on grain kinetics

Grain growth in nanosized polycrystalline Al was studied via molecular dynamics. In particular, the volumetric growth rate of grains as a function of size, topology and mean curvature was analyzed. To this aim, an algorithm for the identification of the topological features and calculation of the mean curvature was developed. It was found that in average grains with ≈15 faces have zero integral mean curvature and show no change in volume implying that the critical number of face (Fcr) is ≈15. Furthermore, integral mean curvature of grains was found to be linearly correlated with the difference between the number of faces grains have and the average number of faces of their neighbors, F−<FNN>. That is, a grain with F−<FNN>=0 tends to have zero integral mean curvature and neither shrinks nor grows whereas grains with F−<FNN> greater than 0 grow and those with F−<FNN> less than 0 shrink. This agrees with the theoretical expectations from MacPherson–Srolovitz relationship and topology based descriptions of grain growth. Nonetheless, considering individual grain kinetics, the comparison of theoretical predictions (MacPherson–Srolovitz relationship, F−<FNN> and topological class) with measured growth rates showed more scatter. In addition, the deviation between the models and the measure values occurred for grains having a substantial relative size difference with the average grain size (|Rgrain−<R>|) and grains with relatively high |F−Fcr|. These grains also had average turning angles at triple lines (βTJ), which differed from the average equilibrium value expected in isotropic metals, π/3, but were consistent with grain growth under finite triple line mobility. This seems to confirm the existence of triple junction drag at the nanocrystalline length scale, which is not accounted for in classical theoretical models.

Effects of peculiar velocities on the morphological properties of large-scale structure

It is known that the large-scale structure (LSS) mapped by a galaxy redshift survey is subject to distortions by galaxies' peculiar velocities. Besides the signatures generated in common N-point statistics, such as the anisotropy in the galaxy 2-point correlation function, the peculiar velocities also induce distinct features in LSS's morphological properties, which are fully described by four Minkowski functionals (MFs), i.e., the volume, surface area, integrated mean curvature and Euler characteristic (or genus). In this work, by using large suite of N-body simulations, we present and analyze these important features in the MFs of LSS on both (quasi-)linear and non-linear scales, with a focus on the latter. We also find the MFs can give competitive constraints on cosmological parameters compared to the power spectrum, probablly due to the non-linear information contained. For galaxy number density similar to the DESI BGS galaxies, the constraint on $\sigma_8$ from the MFs with one smoothing scale can be better by $\sim 50\%$ than from the power spectrum. These findings are important for the cosmological applications of MFs of LSS, and probablly open up a new avenue for studying the peculiar velocity field itself.

Chemically Induced Evolution of Morphological and Connectivity Characteristics of Pore Space of Complex Carbonate Rock via Digital Core Analysis

AbstractWe study the impact of reactive flow on a complex carbonate rock whose pore space was resolved by X‐ray μ‐CT. A small core plug was imaged in the original and chemically altered states. The morphology of the resolved pore space was quantified using various approaches, such as sample‐spanning cluster analysis, capillary drainage transform (CDT), and Minkowski functionals. In particular, we carefully evaluated the changes in macropore connectivity by calculating the fraction of percolated cluster, the Euler characteristic of macroporous phase, and the critical pore diameter in pre‐ and post‐alteration states. We observed that the reactive flow mainly affected larger pores in the macropore region of the sample. More importantly, a huge increase in macropore connectivity, as indicated majorly by the CDT measurements and the Euler characteristic, was observed such that the poorly connected original macropore phase was converted to well‐connected ones after the reactive flow. In addition, the increases in the macropore surface area and integral mean curvature indicated that the oomoldic and interooidal pores of the rock were dominantly enlarged by the reactive fluid. Based on direct numerical simulations on the main flow paths (i.e., the macro‐connected pore phase), strong correlations were found between changes in transport properties and the evolution of connectivity and length scales of the macropore phase, such that simple power‐law models could be used to predict with acceptable accuracy. This suggests that the evolution of transport properties of the rock was mainly controlled by the changes in the macro‐connected porosity and the critical pore diameter.

Integral mean curvature per unit volume and the grain boundary sweep constant

A microstructural property new in application to grain growth, the integral mean curvature per unit volume, MVSgb, has been found to describe the rate of volume sweeping by boundaries throughout the entire structure. It is shown that MVSgb is experimentally obtainable via the sweeping tangent count or from summing the values of MS of all grains within a volume. Measurements on real materials and from computer simulations in the self-similar grain growth regime show that the values of MVSgb and V¯−2/3, where V¯ is the mean grain volume, are essentially equal, though the basis of this experimentally observed similarity remains to be derived. The Rhines/Doherty grain boundary sweep constant, θ*, describing the number fraction of grains lost per volume fraction swept by boundaries, has also been determined for the first time for a 3D grain structure as θ* ≈ −2, constant throughout steady-state grain growth. Combining MVSgb and θ* derives the classic grain growth law expressed as V¯2/3−V¯02/3 = −4/3k(t−t0). Testing shows that the value of the rate constant k from this relationship independently matches the slope k of the plot of dV/dt vs. MS, as theoretically expected. This is the first known time the classic grain growth law of D¯2 ≈ kt, where D¯ is equivalent to the mean linear grain intercept, has been derived using a comprehensive microstructural approach, encompassing the entire structure.

Relationship between mean grain face curvature and number of faces in normal grain growth: The meaning of normalized integral mean curvature

This paper describes the physical meaning of MS norm (=MS/V1/3), the unitless grain size normalized integral mean curvature of grain faces, as a descriptor of the effect of face number on grain shape, affecting grain boundary velocity in curvature driven grain growth. MS norm has consistently been found to have a linear relationship with face class, f, among diverse experiments with real materials and simulations. Analysis of MS norm in terms of the effects of grain size and number of faces on its components, the average mean curvature H¯ and surface area S, show that combined, they reduce to a unitless term that is a function only of grain shape. Values of H¯, approximated as MS /V2/3 are shown for the first time as a function of f and grain scale. Plots of H¯ versus f show positive values at low face classes, decreasing rapidly towards zero at f∼15, and becoming increasingly negative at a decreasing rate with increasing f. Plotted versus linear grain scale, V1/3, the value of H¯ for each face class decreases in magnitude with increased scale during growth. Plots of Ms vs. face class show a continuous decrease as with H¯. Independent plots of Ms vs. linear grain scale for individual face classes were found to be linear through the origin, with slopes equal to MS norm.

Weighted Isoperimetric Inequalities in Warped Product Manifolds

We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities in space forms.

Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov–Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the k k th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when k k is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.

Integral mean curvature analysis of 3D grain growth: Linearity of dV/dt and grain volume

The volumetric growth rate of individual grains has been found to be directly proportional to the individual grain volume V as dV/dt = β(Vo-V). This simple result is explained through the DeHoff relationship dV/dt = −kMs between growth rate and the integral mean curvature of individual grains, Ms, combined with the experimentally observed relationship between Ms and grain volume, Ms = α(Vo-V). These relationships have now been observed consistently in both 3D grain growth simulations and experiments. This paper describes the relationships among these kinetic and geometric grain characteristics that provide this simple description of 3D grain growth.

The role of Minkowski functionals in the thermodynamics of two-phase systems

Within this work quite old concepts from integral geometry are applied to classical equilibrium thermodynamics of two-phase systems. In addition to the area as basic interfacial quantity the full geometric characterization of the interface is used, which includes the two remaining Minkowski functionals, the mean curvature integral and the Euler Poincaré characteristic. The basic energetic characteristic of the interface (i.e. the interfacial tension) is extended by two additional properties: edge force as (up to a factor 4/π) the work necessary to form a right-angled edge of unit length, and item energy as the work to form an additional item in the phase morphology. Both quantities are of increasing importance, when going to micro- and nano-scales. They are subsequently used for interfaces of arbitrary shape to derive a relationship extending the classical Young-Laplace equation. The supplementary contribution is proportional to the Gaussian curvature, with the edge force as proportionality constant. Furthermore, both edge force and item energy are shown to be applicable to the description of crystal nucleation in liquids (extending the classical Becker Döring theory). It turns out, that even above the thermodynamic melting temperature stable nuclei can be present in the liquid phase. They immediately are able to grow when quenched to a temperature below a characteristic temperature. This temperature of spontaneous homogeneous nucleation is simply connected to the edge force, whereas the number of stable clusters per unit volume is dominated by the item energy. Finally, the additional energetic interfacial properties are used in a similar way to characterize the stability of emulsions.

Quantitative imaging: Correlating image features with the segmentation accuracy of PET based tumor contours in the lung

The purpose of this study was to investigate the correlation between image features extracted from PET images and the accuracy of manually drawn lesion contours in the lung. Such correlations are interesting in that they could potentially be used in predictive models to help guide physician contouring. In this work, 26 synthetic PET datasets were created using an anthropomorphic phantom and Monte Carlo simulation. Manual contours of simulated lesions were provided by 10 physicians. Contour accuracy was quantified using five commonly used similarity metrics which were then correlated with several features extracted from the images. Features were sub-divided into three groups using intensity, geometry, and texture as categorical descriptors. When averaged among the participants, the results showed relatively strong correlations with complexity and contrastI (r≥0.65, p<0.001), and moderate correlations with several other image features (r≥0.5, p<0.01). The predictive nature of these correlations was improved through stepwise regression and the creation of multi-feature models. Imaging features were also correlated with the standard deviation of contouring error in order to investigate inter-observer variability. Several features were consistently identified as influential including integral of mean curvature and complexity. These relationships further the understanding as to what causes variation in the contouring of PET positive lesions.

Brendle’s Inequality on Static Manifolds

We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chru\'{s}ciel and Simon \cite{CS}.

Pore scale rock typing and upscaling

Rock-typing is an important part for micro pore scale flow simulation and macro scale reservoir simulation. It is the linkage between pore geometries and fluid flow properties. Recent research advances on pore scale flow simulation enable rock properties to be calculated based on the digital images. But the calculation results doesn’t reflect the properties of whole rock sample. Therefore a suitable upscaling method is needed. In this study we first illustrate three different rock types: depositional rock type, petrographic rock type and hydraulic rock type. Depositional rock type is defined within the context of the large-scale geologic framework and represent those original rock properties present at deposition. Petrographic rock type is also described in the context of the geological framework established from the depositional rock types, but is based on a pore-scale microscopic imaging. Hydraulic rock types is also quantified on the pore scale but represent the physical rock flow and storage properties as controlled by the pore structure. Then we introduce a work flow process that accurately solves the problem of rock typing and upscaling. The reservoir rock is a thin-bedded sandstone from an aeolian environment. For petrographic rock typing, we suppose thin-bedded rock sample as a layer system and define a measurement window moving along the z -axis of the core with Δ z =1 voxel, perpendicular to the bedding direction. Minkowski functions namely total volume, surface area, integral of mean curvature, and integral of Gaussian curvature, are calculated based on the measure windows and kmeans clustering is applied to detect layer interface. Furthermore, the integration of spatial interpolation and kmeans clustering is proved to detect the dipping interface successfully. The classification results match well with corresponding properties and stratigraphic pitch-out can be found, which reflects the aeolian process, e.g., the activity of the winds to enable sediment transport. For hydraulic rock typing, the size, geometry, and distribution of pore throats, as determined by capillary pressure measurements, control the magnitude of porosity and permeability for a given rock. Therefore dynamic geometrical properties during the physical rock flow should be considered and we use percolation test data which includes critical diameter and Minkowski functions. All of these data are calculated based on the pore throats, which the fluid flows on it. The key to successful and efficient characterization is to exploit natural scales. Values and variability of each property are function of scale due to heterogeneities. Therefore, representative elementary volumes should be selected suitably to stabilize the data measurements. We first choose different sizes of subsample and measure corresponding percolation test data, permeability and effective conductivity. After selecting suitable representative elementary volumes, bottom blocks of subsamples are used for unsupervised Gaussian mixture model clustering and top blocks of subsamples are used for supervised Gaussian mixture model clustering based on the calculated centroid which means the mean value and covariance matrix of the percolation test data for each rock type in bottom blocks. The results demonstrate that different rock types are well separated with each other and match well with the corresponding geometrical properties and petrophysical properties. Three formulas which are based on layer system, undulating surfaces and hydraulic rock typing results, are developed to do upscaling. Local permeabilities and effective conductivities are computed to predict whole rock sample’s permeabilities and effective conductivities. The results indicate that upscaling formula based on the hydraulic rock typing results is most accurate.

Gauss–Bonnet formulae and rotational integrals in constant curvature spaces

We obtain generalizations of the main result in [10], and then provide geometric interpretations of linear combinations of the mean curvature integrals that appear in the Gauss–Bonnet formula for hypersurfaces in space forms Mλn. Then, we combine these results with classical Morse theory to obtain new rotational integral formulae for the k-th mean curvature integrals of a hypersurface in Mλn.

Local Digital Algorithms Applied to Boolean Models

AbstractWe investigate the estimation of specific intrinsic volumes of stationary Boolean models by local digital algorithms; that is, by weighted sums of local configuration counts. We show that asymptotically unbiased estimators for the specific surface area or integrated mean curvature do not exist if the dimension is at least two or three, respectively. For three‐dimensional stationary isotropic Boolean models, we derive asymptotically unbiased estimators for the specific surface area and integrated mean curvature. For a Boolean model with balls as grains, we even obtain an asymptotically unbiased estimator for the specific Euler characteristic.