Manifolds Of Codimension Research Articles

𝐶^{1,1-𝜖} isometric embeddings

In this paper we use the convex integration technique enhanced by an extra iteration originally due to Källén and revisited by Kröner to provide a local h h -principle for isometric embeddings in the class C 1 , 1 − ε C^{1,1-\varepsilon } for n n -dimensional manifolds in codimension 1 2 n ( n + 1 ) \frac {1}{2}n(n+1) .

Iso-contact embeddings of manifolds in co-dimension $2$

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$ provided $M$ contact embeds in $(N, \xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $\xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.

Polytopic discontinuous Galerkin methods for the numerical modelling of flow in porous media with networks of intersecting fractures

We present a numerical approximation of Darcy's flow through a porous medium that incorporates networks of fractures with non empty intersection. Our scheme employs PolyDG methods, i.e. discontinuous Galerkin methods on general polygonal and polyhedral (polytopic, for short) grids, featuring elements with edges/faces that may be in arbitrary number (potentially unlimited) and whose measure may be arbitrarily small. Our approach is then very well suited to tame the geometrical complexity featured by most of applications in the computational geoscience field. From the modelling point of view, we adopt a reduction strategy that treats fractures as manifolds of codimension one and we employ the primal version of Darcy's law to describe the flow in both the bulk and in the fracture network. In addition, some physically consistent conditions couple the two problems, allowing for jump of pressure at their interface, and they as well prescribe the behaviour of the fluid along the intersections, imposing pressure continuity and flux conservation. Both the bulk and fracture discretizations are obtained employing the Symmetric Interior Penalty DG method extended to the polytopic setting. The key instrument to obtain a polyDG approximation of the problem in the fracture network is the generalization of the concepts of jump and average at the intersection, so that the contribution from all the fractures is taken into account. We prove the well-posedness of the discrete formulation and perform an error analysis obtaining a priori hp-error estimates. All our theoretical results are validated performing preliminary numerical tests with known analytical solution.

Chaotic attractors in Atkinson–Allen model of four competing species

We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

Singular Schrödinger operators and Robin billiards

This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schrodinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.

Integration over curves and surfaces defined by the closest point mapping

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation—initially derived to integrate over manifolds of codimension one—to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.

Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

We consider the energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$. For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} + \frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$ . Given an integer $L\in \Bbb N^*$, we exhibit a set of initial data arbitrarily close to the least energy harmonic map $Q$ in the energy critical topology such that the corresponding solution blows up in finite time by concentrating its energy $$\nabla u(t,r)-\nabla Q(\frac{r}{\l(t)})\to u^* in L^2$$ at a speed given by the {\it quantized} rates: $$\l(t)=c(u_0)(1+o(1))\frac{(T-t)^L}{|\log (T-t)|^{\frac{2L}{2L-1}}},$$ in accordance with the formal predictions [3]. The case L=1 corresponds to the stable regime exhibited in [37], and the data for $L\ge 2$ leave on a manifold of codimension $(L-1)$ in some weak sense. Our analysis lies in the continuation of [36,32,37] by further exhibiting the mechanism for the existence of the excited slow blow up rates and the associated instability of these threshold dynamics.

Attractors–Repellers in the Space of Contours in the Stokes–Leibenson Problem for Hele–Shaw Flows

It is shown that, in the space of quasicontours playing a role of free boundaries in the Stokes–Leibenson problem, there is a manifold of codimension 1 such that some points of this manifold are attractors in the case of sink and repellers in the case of source, whereas, on the contrary, other points are repellers in the case of sink and attractor in the case of source. Bibliography: 36 titles. Illustrations: 7 figures.

Regularizing Piecewise Smooth Differential Systems: Co-Dimension $$2$$ Discontinuity Surface

In this paper, we are concerned with numerical solution of piecewise smooth initial value problems. Of specific interest is the case when the discontinuities occur on a smooth manifold of co-dimension 2, intersection of two co-dimension 1 singularity surfaces, and which is nodally attractive for nearby dynamics. In this case of a co-dimension 2 attracting sliding surface, we will give some results relative to two prototypical time and space regularizations. We will show that, unlike the case of co-dimension 1 discontinuity surface, in the case of co-dimension 2 discontinuity surface the behavior of the regularized problems is strikingly different. On the one hand, the time regularization approach will not select a unique sliding mode on the discontinuity surface, thus maintaining the general ambiguity of how to select a Filippov vector field in this case. On the other hand, the proposed space regularization approach is not ambiguous, and there will always be a unique solution associated to the regularized vector field, which will remain close to the original co-dimension 2 surface. We will further clarify the limiting behavior (as the regularization parameter goes to 0) of the proposed space regularization to the solution associated to the sliding vector field of Dieci and Lopez (Numer Math 117:779–811, 2011). Numerical examples will be given to illustrate the different cases and to provide some preliminary exploration in the case of co-dimension 3 discontinuity surface.

Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field

In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p ≥ 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict to cases where the sliding manifold is attractive. For the case of co-dimension p = 2, we will distinguish between two types of attractive sliding manifold: “node-like” and “spiral-like”. The case of node-like attractive manifold will be further extended to the case of p ≥ 3. Finally, we compare our model to other existing methodologies on some examples.

Asymptotic behavior of eigenvalues of Schrödinger type operators with degenerate kinetic energy

AbstractWe study the eigenvalues of Schrödinger type operators T + λV and their asymptotic behavior in the small coupling limit λ → 0, in the case where the symbol of the kinetic energy, T (p), strongly degenerates on a non‐trivial manifold of codimension one (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Discretizing delta functions via finite differences and gradient normalization

In [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915–931] the author presented two closely related finite difference methods (referred to here as FDM1 and FDM2) for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u:Rn↦R. These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h→0) in the codimension one setting.In this paper, we concentrate on n⩽3, but generalize our methods to codimensions other than one – now the level set function is generally a vector valued mapping u→:Rn↦Rm,1⩽m⩽n⩽3. Seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems. Motivated by this, we instead use the wedge product formalism to generalize our FDM algorithms, and this approach results in accurate, often consistent approximations. With the goal of ensuring consistency in general, we propose a new gradient normalization process that is applied before our FDM algorithms. These combined algorithms seem to be consistent in all reasonable situations, with numerical experiments indicating O(h2) convergence for our new gradient-normalized FDM2 algorithm.In the full codimension setting (m=n), our gradient normalization processing also improves accuracy when using more standard approximate delta functions. This combination also yields approximations that appear to be consistent.

Finite difference methods for approximating Heaviside functions

We present a finite difference method for discretizing a Heaviside function H ( u ( x → ) ) , where u is a level set function u : R n ↦ R that is positive on a bounded region Ω ⊂ R n . There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915–931; J.D. Towers, Discretizing delta functions via finite differences and gradient normalization, Preprint at http://www.miracosta.edu/home/jtowers/; J.D. Towers, A convergence rate theorem for finite difference approximations to delta functions, J. Comput. Phys. 227 (2008) 6591–6597]. We consider our approximate Heaviside functions as they are used to approximate integrals over Ω . We prove that our first approximate Heaviside function leads to second order accurate quadrature algorithms. Numerical experiments verify this second order accuracy. For our second algorithm, numerical experiments indicate at least third order accuracy if the integrand f and ∂ Ω are sufficiently smooth. Numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations. We mostly focus on smooth f and u. By this we mean that f is smooth in a neighborhood of Ω , u is smooth in a neighborhood of ∂ Ω , and the level set u ( x ) = 0 is a manifold of codimension one. However, our algorithms still give reasonable results if either f or u has jumps in its derivatives. Numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way, assuming that the level set u ( x ) = 0 is a manifold. Numerical experiments indicate that dependence on the placement of Ω with respect to the grid is quite small for our algorithms. Specifically, a grid shift results in an O ( h p ) change in the computed solution, where p is the observed rate of convergence.

On codimensions k immersions of m-manifolds for k = 1 and k = m − 2

Let us consider M a closed smooth connected m-manifold, N a smooth (2m − 2)-manifold and \({ f : M \longrightarrow N}\) a continuous map, with \({ m \equiv 1(4)}\). We prove that if \({ {f}_* : {H}_{1}(M; \, Z_2) \longrightarrow \check{H}_{1}(f(M) ; \, Z_2)}\) is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97–110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281–285, 1992).

The compression theorem III: applications

This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with applications. We give short new (and constructive) proofs for immersion theory and for the loops--suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions. We also consider the general problem of controlling the singularities of a smooth projection up to C^0--small isotopy and give a theoretical solution in the codimension > 0 case.

Modulation Equations of Weakly Nonlinear Geometrical Optics in Media Exhibiting Mixed Nonlinearity

The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propagating in materials exhibiting mixed nonlinearity is derived with the resonant wave interactions inherent in the system. Our analysis differs from the results of Hunter et al., in that we have employed a different scaling, keeping in view the delayed effects of nonlinearity in certain thermodynamic systems exhibiting mixed nonlinearity. The result is to modify the transport equations obtained by Hunter et al. by the addition of certain cubic nonlinear terms. Through the method of averaging, the secular terms are eliminated. However, the averaging process is carried out in two steps; first, along manifolds of codimension two giving an advection equation, the solution of which is then averaged in a direction transverse to the above‐mentioned manifold.

Critical sets of nonlinear Sturm-Liouville operators of Ambrosetti-Prodi type

The critical set C of the operator F:X→L2([0,π]) defined byF(u) = -u'' + f(u) is studied. Here X: = H2D([0,π]) stands for the setof functions that satisfy the Dirichlet boundary conditions and whosederivatives are in L2([0,π]). For generic nonlinearities f,C = ∪Ck decomposes into manifolds of codimension 1 in X. Iff''<0 or f''>0, the set Cj is shown to be nonempty if, and onlyif, -j2 (the jth eigenvalue of u'') is in the range of f'. Thecritical components Ck are (topological) hyperplanes.

Sufficient conditions for relative minima of broken extremals in optimal control theory

We formulate a version of the method of characteristics based on parametrizations of extremals by their terminal values. Sufficient conditions are given for imbedding a reference trajectory into a local field of broken extremals. For a problem with terminal manifold of codimension 1 it is shown that a broken extremal is a relative minimum if (i) the restrictions of the flow to intervals where the control is continuous have nonsingular partial derivatives with respect to the parameter and (ii) the switching surfaces are crossed transversally.

Unique continuation for elliptic operators with non-multiple characteristics

In this paper we prove a unique continuation theorem for elliptic operators of the formP(D)+V(x), whereP(D) has orderm≥2 and simple complex characteristics, andV(x)∈L n/m (R n ). To prove our main theorem we use a restriction theorem for the Fourier transform to manifolds of codimension 2.

Billiards in Tubular Neighborhoods of Manifolds¶of Codimension 1

I prove that in (sufficiently small) tubular ρ$-neighborhoods of a given C 3 manifold of codimension 1, any two points can be connected by a billiard trajectory, and that in addition there exists such a trajectory having at most \(\) collision points, for a suitable H>0, provided the manifold is of class C 3.