The cross ratio and Clifford algebras

Motivated by recent quaternionic approach of Bobenko and Pinkall to the complex cross ratio we presenta simple method to eva]uate the cross ratio in the Euclidean space? ⁿ identifying the space with vectors generating the Clifford algebraC(n). We apply the Clifford cross ratio to describe discrete analogues of orthogonal nets in? ⁿ .     

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \mathbb Sⁿ⁺¹{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.     

Special John-Nirenberg-Campanato spaces via congruent cubes

Let p ∈ [1, ∞), q ∈ [1, ∞), α∈ R, and s be a non-negative integer. Inspired by the space JNp introduced by John and Nirenberg(1961) and the space B introduced by Bourgain et al.(2015), we introduce a special John-Nirenberg-Campanato space JN^con_(p,q,s) over Rⁿ or a given cube of R;with finite side length via congruent subcubes, which are of some amalgam features. The limit space of such spaces as p →∞ is just the Campanato space which coincides with the space BMO(the space of functions with bounded mean oscillations)when α = 0. Moreover, a vanishing subspace of this new space is introduced, and its equivalent characterization is established as well, which is a counterpart of the known characterization for the classical space VMO(the space of functions with vanishing mean oscillations) over Rⁿ or a given cube of Rⁿ with finite side length.Furthermore, some VMO-H¹-BMO-type results for this new space are also obtained, which are based on the aforementioned vanishing subspaces and the Hardy-type space defined via congruent cubes in this article. The geometrical properties of both the Euclidean space via its dyadic system and congruent cubes play a key role in the proofs of all these results.     

Finite Morse Index Implies Finite Ends

We prove that finite Morse index solutions to the Allen-Cahn equation in ℝ² have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second-order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second-order regularity of clustering interfaces in ℝⁿ is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝ^n–1. For finite Morse index solutions in ℝ², we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.     

Clifford algebras and Maxwell's equations in Lipschitz domains

We present a simple, Clifford algebra‐based approach to several key results in the theory of Maxwell's equations in non‐smooth subdomains of ℝ^m. Among other things, we give new proofs to the boundary energy estimates of Rellich type for Maxwell's equations in Lipschitz domains from [20, 10], discuss radiation conditions and the case of variable wave number. Copyright © 1999 John Wiley & Sons, Ltd.     

Sufficient conditions for second order L2-convergence of the fractional step theta method

We consider the fractional step theta time stepping procedure for the non-stationary incompressible linear Stokes equations in a cylindrical domain (0, T) × Ω, where Ω is a bounded domain in ℝⁿ. Using energy estimates and assuming a certain degree of regularity for the data, we show second order L²-convergence. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

The Incompressible Limit and the Initial Layer of the Compressible Euler Equation in ℝn+

We consider the incompressible limit of the compressible Euler equation in the half-space ℝⁿ₊. It is proved that the solutions of the non-dimensionalized compressible Euler equation converge to the solution of the incompressible Euler equation when the Mach number tends to zero. If the initial data v^∞₀ do not satisfy the condition ‘∇⋅v^∞₀=0’, then the initial layer will appear. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On uniqueness of invariant measures for finite‐ and infinite‐dimensional diffusions

We prove uniqueness of “invariant measures,” i.e., solutions to the equation L*μ = 0 where L = Δ + B · ∇ on ℝⁿ with B satisfying some mild integrability conditions and μ being a probability measure on ℝⁿ. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L¹(μ) generates a strongly continuous semigroup having μ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L¹(μ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the “symmetric case”) in particular is studied and conditions are identified ensuring that L*μ = 0 implies that L is symmetric on L²(μ) or L*μ = 0 has a unique solution. We also prove infinite‐dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. © 1999 John Wiley & Sons, Inc.     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

Cohomogeneity one de Sitter space S n 1

In this paper we study the cohomogeneity one de Sitter space S1 n. We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.     

Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ℕ², Ω an open bounded domain in ℝ^m , Y = [0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ℝ^m × ℝⁿ → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝ^m × ℝⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that

     

Orientable and non-orientable director fields for liquid crystals

Uniaxial nematic liquid crystals are often modelled using the Oseen-Frank theory, in which the mean orientation of the rodlike molecules is described through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to −n; that is, instead of n taking values in the unit sphere 𝕊², it should take values in the sphere with opposite points identified, i.e. in the real projective plane ℝP². The Landau-de Gennes theory respects this symmetry by working with the tensor Q = s (n ⊗ n − ⅓Id). In the case of a non-zero constant scalar order parameter s the Landau-de Gennes theory is equivalent to that of Oseen-Frank when the director field is orientable. We report on a general study of when the director fields can be oriented, described in terms of the topology of the domain filled by the liquid crystal, the boundary data and the Sobolev space to which Q belongs. We also analyze the circumstances in which the non-orientable configurations are energetically favoured over the orientable ones. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quadratic forms with values in invertible modules

LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the type 1 subgroup).     

Willmore Lagrangian Spheres in the Complex Euclidean Space C n

In this paper we construct many examples of n-dimensionalWillmore Lagrangian submanifolds in the complex Euclidean space C ⁿ . We characterize them as the only Willmore Lagrangian submanifolds invariant under the action of SO(n). The mostimportant contribution of our construction is that it provides examplesof Willmore Lagrangian spheres in C ⁿ for all n 2.     

Existence and uniqueness results for a nonlinear differential system

We study the existence, uniqueness and asymptotic behavior of the strong and weak solutions of a nonlinear differential system with 2N equations in a real Hilbert space H, subject to a boundary condition and initial data. This problem is a discrete version with respect to spatial variable x of some partial differential problems (with H = ℝⁿ ), which have applications in integrated circuits modelling     

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Abstract

We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.     

On the convergence of stable phase transitions

We consider the local behavior of critical points of the functional as ε → 0. Here, W is a double-well potential and U is a regular domain in ℝⁿ, n ≥ 2. Assuming that {u^ε}_ε>0 is stable for n = 2 and locally energy-minimizing for n = 3, we show that the level sets of solutions converge in an average sense to a stationary (n − 1)-rectifiable varifold. Our study is based on estimates derived from the second variation formula and is entirely local. © 1998 John Wiley & Sons, Inc.     

A Note on a Marcinkiewicz Integral Operator

We prove the L_p boundedness of Marcinkiewicz integral operators with rough kernels on ℝⁿ and T ⁿ.     

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.