<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

<Emphasis Type="Italic">h</Emphasis>-Principles for curves and knots of constant curvature

We prove that curves of constant curvature satisfy, in the sense of Gromov, the relative -dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given knot f there exists a knot f͂ of constant curvature which is -close to f. More importantly, we show that if f is , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull. The author’s research was supported in part by NSF CAREER award DMS-0332333.     

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B _n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e ₁ , ..., e _n denoting the standard basis vectors of ⁿ and letting x _k = e ₁ + ··· + e _k (k = 1, 2, ..., n), the set

A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Let be a Minkowski 3-space of Randers type with , where is the Euclidean metric and . We consider minimal surfaces in and prove that if a connected surface M in is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of , M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x ³-axis.      

Constant mean curvature graphs in a class of warped product spaces

We introduce a new existence result for compact normal geodesic graphs with constant mean curvature and boundary in a class of warped product spaces. In particular, our result includes that of normal geodesic graphs with constant mean curvature in hyperbolic space over a bounded domain in a totally geodesic .      

Uniform Schauder estimates for regularized hypoelliptic equations

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: in , where Δ is the Laplace operator, m < n, and the limit operator is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter ϵ, of solution of the approximated equation L _ϵ u = f, using a modification of the lifting technique of Rothschild and Stein. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.      

Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation

Rectifiable oscillations in second-order half-linear differential equations

Second-order half-linear differential equation (H): on the finite interval I = (0,1] will be studied, where , p > 1 and the coefficient f(x) > 0 on I, , and . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when , λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L ^p nonintegrability of the derivative of all solutions of the equation (H).      

Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).     

Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in \mathbb{C}^n

We give new examples of self-shrinking and self-expanding Lagrangian solutions to the Mean Curvature Flow (MCF). These are Lagrangian submanifolds in , which are foliated by (n − 1)-spheres (or more generally by minimal (n − 1)-Legendrian submanifolds of ), and for which the study of the self-similar equation reduces to solving a non-linear Ordinary Differential Equation (ODE). In the self-shrinking case, we get a family of submanifolds generalising in some sense the self-shrinking curves found by Abresch and Langer.     

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.      

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .      

The norm of products of free random variables

Let X _i denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X _i it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX _i = 1, then is between and c ₂ n for certain constant c ₁ and c ₂. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X _i , and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables.     

Renormalized solutions of nonlinear parabolic equations with general measure data

Let a bounded open set, N ≥  2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is

Asymptotic symmetries for conformal scalar curvature equations with singularity

We give conditions on a positive Hölder continuous function C²such that every C ² positive solution u((x)) of the conformal scalar curvature equation in a punctured neighborhood of the origin in R ⁿ either has a removable singularity at the origin or satisfies for some positive singular solution u ₀(|x|) of where is the Hölder exponent of K.Mathematics Subject Classification (2000) Primary 35J60, 53C21     

Connecting rational homotopy type singularities

Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W ^1,p (R ^p+1, N) be the Sobolev space of measurable maps from R ^p+1 into N whose gradients are in L ^p . The restriction of u to almost every p-dimensional sphere S in R ^p+1 is in W ^1,p (S, N) and defines an homotopy class in π _p (N) (White 1988). Evaluating a fixed element z of Hom(π _p (N), R) on this homotopy class thus gives a real number Φ _z,u (S). The main result of the paper is that any W ^1,p -weakly convergent limit u of a sequence of smooth maps in C ^∞(R ^p+1, N), Φ _z,u has a rectifiable Poincaré dual . Here Γ is a a countable union of C ¹ curves in R ^p+1 with Hausdorff -measurable orientation and density function θ: Γ→R. The intersection number between and S evaluates Φ _z,u (S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n _z , depending only on homotopy operation z, such that even though the mass may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n _z ) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans” which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R ^m for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented Euclidean p-sphere.