Existence of Constant Mean Curvature Disks in $$\mathbb {R}^3$$ with Capillary Boundary Condition

We show that some pieces of cylinders bounded by two parallel straight lines bifurcate in a family of periodic nonrotational surfaces with constant mean curvature and with the same boundary conditions. These cylinders are initial interfaces in a problem of microscale range modeling the morphologies that adopt a liquid deposited in a chemically structured substrate with striped geometry or a liquid contained in a right wedge with Dirichlet and capillary boundary condition on the edges of the wedge. Experiments show that starting from these cylinders and once a certain stage is reached, the shape of the liquid changes drastically in an abrupt manner. Studying the stability of such cylinders, the paper provides a mathematical proof of the existence of these new interfaces obtained in experiments. The analysis is based on the theory of bifurcation by simple eigenvalues of Crandall and Rabinowitz.

We prove ‐regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free‐boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half‐plane with angle not equal to , then it coincides with a properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.

Numerical simulators developed to match coreflooding performance provide a powerful means for the determination of relative permeabilities of a core sample. However, the application of proper capillary boundary conditions at the ends of a sample is not straightforward. In this paper, the impact of inlet boundary conditions on the numerical simulation of one-dimensional coreflooding is studied. It is found that (i) the widely used inlet saturation boundary condition - the inlet saturation rises instantly to its maximum value - is not correct, and instead the inlet saturation increases gradually depending on the capillary number; (ii) if the maximum inlet saturation condition is applied, numerical solutions would not satisfy both the frontal advance equation and the fractional flow equation; and (iii) the pressure drop history is directly related to the inlet saturation. Hence, a variable inlet saturation boundary condition must be used in the simulation to account for inlet capillary end effects. Thereafter, the pressure drop history (including initial responses) can be used to make performance matches. Introduction Numerical simulations of immiscible corefloods have been widely used to study two-phase flows in a core sample including capillary and gravity force effects(1,2,3,4.5.6.7), and to interpret the results of unsteady state coreflooding experiments-history matching to determine relative permeabilities and capillary pressure(8,9,10). In coreflooding experiments, capillary end effects always exist and can greatly affect displacement performance(11,12,13,14,15.16,17). Although some experimental techniques have been developed to eliminate these effects(18,19), they may still exist and affect displacement performance, especially when reservoir fluids and flow parameters are utilized. To account for the capillary end effects, proper boundary or initial conditions must be applied at the core ends in numerical simulations. The outlet end capillary effect has been incorporated into the simulations performed by Fassihi(10). Recently, Shen and Ruth(7) derived a set of initial and boundary conditions which can properly describe the capillary effect on the variation of the inlet saturation and applied them in their simulations. The application of proper capillary boundary conditions at the ends of a sample in simulations is not straight forward. In spite of long awareness of the inlet end effect on coreflooding flow (11,12,13,15,17), previous numerical simulations(1,2.3,4,5,6) usually could not account for it, with the result of errors in the solutions. It can be anticipated that the solution with proper initial and boundary conditions will be self-consistent and the solution with improper ones will not be. Therefore, the theme of the present study is to investigate the impact of the inlet end capillary effect on the results of numerical simulations. In the subsequent part of the paper, the notion of an inlet end effect indicates the inlet end capillary effect on the variation of the inlet saturation. The numerical simulation technique employed in this study is the same one as used in the previous study(7). The finite element method was employed to obtain dynamic solutions of a Lagrangian equation derived by Bentsen(6) (Bentsen's equation). This paper consists of the following parts: Firstly, a brief summary of the derivation of Bentsen's equation is presented.

One of the most important results in differential geometry is that the only closed hypersurfaces of constant mean curvature and in general constant higher order mean curvature) embedded in Euclidean space are round spheres [1]. This result is not true for the case of immersed (and non-embedded hypersurfaces [11, 14]. Many generalizations of this result have been obtained later, for example constant scalar curvatures or constant higher order mean curvatures hypersurfaces [2,3,7,9]. As a natural generalization of hypersurfaces with constant mean curvature or with constant higher order mean curvature, linear Weingarten and more general generalized, Weingarten hypersurfaces hypersurface has been studied in many places. [5],[10]. The aim of our work is to establish a characterization theorem concerning complete generalized Weingarten hypersurfaces embedded in Euclidean space. That is an hypersurfaces where some of the higher order mean curvature are linearly related. We prove that the only closed, oriented generalized Weingarten hypersurfaces embedded in Euclidean space with non-vanishing higher order mean curvature are the round spheres. This result generalizes the cases of constant higher order mean curvature hypersurfaces and linear Weingarten hypersurfaces embedded in Euclidean space.

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M 5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E 5 having constant higher order mean curvature Hr for r > 2.

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds). As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C k C^{k} -norms. Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres.

We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

In this paper, we study a Serrin-type anisotropic overdetermined problem with anisotropic capillary boundary in a half-space R + n \mathbb {R}^{n}_{+} . We prove that a domain on which this partially anisotropic overdetermined problem admits a solution if and only if the domain is a truncated Wulff ball intersecting ∂ R + n \partial \mathbb {R}^{n}_{+} with constant anisotropic capillary angle, which generalizes the isotropic results of Jia-Lu-Xia-Zhang [Calc. Var. Partial Differential Equations 63 (2024), 23 pp.].

Spacelike hypersurfaces with constant higher order mean curvature in Minkowski space–time

An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, and it is a generalization of surface area. Equilibrium surfaces with volume constraint are called CAMC (constant anisotropic mean curvature) surfaces and they are not smooth in general. We show that, if the energy density function is two times continuously differentiable and convex, then, like isotropic (constant mean curvature) case, the uniqueness for closed stable CAMC surfaces holds under the assumption of the integrability of the anisotropic principal curvatures. Moreover, we show that, unlike the isotropic case, uniqueness of closed embedded CAMC surfaces with genus zero in the three-dimensional euclidean space does not hold in general. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the Euclidean space with general dimension. This article is an announcement of two forthcoming papers by the author.

Biconservative submanifolds, with important role in mathematical physics and differential geometry, arise as the conservative stress-energy tensor associated to the variational problem of biharmonic submanifolds. Many examples of biconservative hypersurfaces have constant mean curvature. A famous conjecture of Bang-Yen Chen on Euclidean spaces says that everybiharmonic submanifold has null mean curvature. Inspired by Chen conjecture, we study biconservative Lorentz submanifolds of the Minkowski spaces. Although the conjecture has not been generally confirmed, it has been proven in many cases, and this has led to its spread to various types of submenifolds. As an extension, we consider a advanced version of the conjecture (namely, $L_1$-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean space $\mathbb{M}^5 :=\mathbb{E}^5_1$ (i.e. the Minkowski 5-space). We show every $L_1$-biconservative Lorentz hypersurface of $\mathbb{M}^5$ with constant mean curvature and at least three principal curvatures has constant second mean curvature.

As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean\ncurvature implies a homological conservation law for hypersurfaces in ambient\nspaces with Killing fields.In Theorem 3.5 here, we generalize that law by\nrelaxing the topological restrictions assumed in [KKS] and by allowing a\nweighted mean curvature functional. We also prove a partial converse (Theorem\n4.1) which roughly says that when flux is conserved along a Killing field, a\nhypersurface splits into two regions: one with constant (weighted) mean\ncurvature, and one preserved by the Killing field. We demonstrate our theory by\nusing it to derive a first integral for helicoidal surfaces of constant mean\ncurvature in Euclidean 3-space, i.e., "twizzlers."\n

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of Rn which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n − 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n − 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.