Conformal Killing Graphs with Prescribed Mean Curvature

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.     

Radial Graphs of Constant Mean Curvature and Doubly Connected Minimal Surfaces with Prescribed Boundary

In this paper we investigate existence and uniqueness of radial graphs ofconstant mean curvature (cmc) with prescribed boundary. Our main resultestablishes the existence of a minimal radial anullus spanning two givenconvex curves in parallel planes of R³; we also obtain a variant ofa well-known result of James Serrin about the existence of radial cmc graphsover convex domains in the sphere.     

Geodesic Graphs with Constant Mean Curvature in Spheres

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere of radius a. Given a compact domain , with some conditions, contained in a totally geodesic sphere S of and a real differentiable function on , we define the graph of in considering the height (x) on the minimizing geodesic joining the point x of to a fixed pole of . For a real number H such that |H| is bounded for a constant depending on the mean curvature of the boundary of and on a fixed number in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary , whenever the diameter of is smaller than a constant depending on . If we have conditions on , that is, let be a graph over of a function, if |H| is bounded for a constant depending only on the mean curvature of and if the diameter of is smaller than a constant depending on H and , then we prove that there exists a unique graphs with mean curvature H and boundary . The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.     

Starshaped Compact Hypersurfaces with Prescribed M-th Mean Curvature in Elliptic Space

We consider the problem of finding a compact starshaped hypresurface in a space form for which the normalized m-th elementary symmetric function of principal curvatures is a prescribed function. In this paper the conditions for the existence of at least one solution to a nonlinear second order elliptic equation of that problem are established in case of a space form with positive sectional curvature.     

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface Σ is a graph in ?⁴ if there is a unit constant 2-form ω on ?⁴ such that <e ₁∧e ₂, ω≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if $ \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} A surface Σ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that <e ₁∧e ₂, ω>≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if v ₀≥ on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M ₁×M ₂ where M ₁ and M ₂ are Riemann surfaces, if <e ₁∧e ₂, ω₁>≥v ₀>0 where ω₁ is a K?hler form on M ₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, v ₀≥ on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic. Received July 25, 2001, Accepted October 11, 2001     

A Note on Doubly Connected Surfaces of Constant Mean Curvature with Prescribed Boundary

In this paper we prove the existence of a constant mean curvature surface spanning two given convex curves in parallel planes of ℝ³ under hypotheses relating the distance between the planes, the curvature of the curves and the mean curvature. It is also proved that the surface is a radial graph over a unit sphere.This author was partially Supported by Fapergs. Mathematics Subject Classifications (2000): 53A10, 53C42, 35J60     

Spacelike Graphs with Parallel Mean Curvature in Pseudo-Riemannian Product Manifolds

The author introduces the w-function defined on the considered spacelike graph M.Under the growth conditions w = o(log z) and w = o(r),two Bernstein type theorems for M in Rmn+ mare got,where z and r are the pseudo-Euclidean distance and the distance function on M to some fixed point respectively.As the ambient space is a curved pseudoRiemannian product of two Riemannian manifolds(Σ1,g1) and(Σ2,g2) of dimensions n and m,a Bernstein type result for n = 2 under some curvature conditions on Σ1 and Σ2 and the growth condition w = o(r) is also got.As more general cases,under some curvature conditions on the ambient space and the growth condition w = o(r) or w = o(√r),the author concludes that if M has parallel mean curvature,then M is maximal.     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

A Barrier Principle for Surfaces with Prescribed Mean Curvature and Arbitrary Codimension

We consider two dimensional surfaces ${X : \Omega\to\mathbb R^{n+2}, \Omega\subset \mathbb C, w=u+iv\mapsto X(w)}$ with arbitrary codimension n and prove a barrier principle for strong (possibly branched) subsolutions ${X\in C^1(\Omega, \mathbb {R}^{n+2})\cap H_{2,{\rm loc}}^2(\Omega,\mathbb R^{n+2})}$ of the integral inequality $$\int_{\Omega} \Big\lbrace \langle \nabla X, \nabla \varphi\rangle +2W \sum_{k=1}^n H_k \langle N_k,\varphi \rangle \Big\rbrace \; dudv\ge 0$$ with mean curvature functions (H _k ) _k=1,...,n which lie locally on one side of a supporting hypersurface S. We show under suitable assumption on the 2-mean curvature of the supporting surface S that X is locally contained in S. This generalizes a corresponding result for surfaces in ${\mathbb R^3}$ , cf. (Dierkes et al., Regularity of Minimal Surfaces, §4.4, 2010).     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Periodic Solutions for a Class of n-dimensional Prescribed Mean Curvature Equations

In this paper,by using an extension of Mawhin’s continuation theorem and some analysis methods,we study the existence of periodic solutions for the following prescribed mean curvature system■where x∈Rⁿ,W∈C¹(Rⁿ,R),p∈C(R,Rⁿ)is T-periodic and■     

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

This paper deals with the following prescribed boundary mean curvature problem in B^N ■ where ■ is a bounded nonnegative function with ■ .Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that N≥5 and ■ has a stable critical point ■ with r₀> d and ■>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may inlude the sad...     

具偏差变元的广义平均曲率方程同宿解的存在性

In this paper, by using Mawhin＇s continuation theorem and some analysis methods,the existence of a set with 2kT-periodic solutions for a kind of prescribed mean curvature equation with a deviating argument is studied, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above set.     

Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

We consider a system of three surfaces, graphs over a bounded domain in ?², intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to 2π/3.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic Hölder spaces), for sufficiently regular initial data satisfying a compatibility condition.     

Mean Curvature,Threshold Dynamics,and Phase Field Theory on Finite Graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as “freezing” or “pinning”) and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.     

有约束条件的图的分数k-因子

设G是一个简单的无向图，若G不是完全图，G的孤立韧度定义为I(G)=min{｜s｜／i(G-S)：S∈V(G)，i(G-S)≥2)；否则令I(G)=∞．对与图的孤立韧度I(G)密切相关的新参数，I’(G)，若G不是完全图，定义I’(G)=min{｜s｜/i(G-S)-1:S∈V(G)，i(G-S)≥2}；否则I’(G)=∞本文研究了新参数I‘(G)与图的分数κ-因子的关系，给出了具有某些约束条件的图的分数κ-因子存在的一些充分条件．     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

Surface in R^3 with Prescribed Gauss Curvature

Assume that a smooth function K(x, y) vanishes on a simple closed smooth curve \Gamma and dK|_\Gamma \ne 0. Then the result of the present paper shows that there is a sufficiently smooth surface z=z(x, y) defined in a neighbourhood of \Gamma with curvature K.     

Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the n-sphere, with n 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existence of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.Mathematics Subject Classification (2000): 35J60, 53C21, 58J05Send offprint requests to: Khalil El Mehdi