Submanifolds Of Rn Research Articles

Multi-Bump Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: The Topological Effect of Potential Wells

Abstract In this article, we are interested in multi-bump solutions of the singularly perturbed problem - ε 2 ⁢ Δ ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) in ⁢ ℝ N . -\varepsilon^{2}\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^{N}. Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki–Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as ε → 0 {\varepsilon\rightarrow 0} . Examples of potential wells include the following: the union of two compact smooth submanifolds of ℝ N {\mathbb{R}^{N}} where these two submanifolds meet at the origin and an embedded topological submanifold of ℝ N {\mathbb{R}^{N}} .

Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

Abstract We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f.

Counting lattice points and weak admissibility of a lattice and its dual

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box. The error term is expressed in terms of a certain function ν(Γ⊥, ·) of the dual lattice Γ⊥, and we carefully analyse the relation of this quantity with ν(Γ, ·). In particular, we show that ν(Γ⊥, ·) = ν(Γ, ·) for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. This result relies on Beresnevich’s recent breakthrough on Davenport’s problem regarding badly approximable points on submanifolds of ℝn. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.

Approximation of generalized ridge functions in high dimensions

This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of RN. We introduce the notion of linear-sleeve functions, whose function values only depend on the distance to some unknown linear subspace L. We propose two effective algorithms to approximate linear-sleeve functions f(x)=g(dist(x,L)2), when both the linear subspace L⊂RN and the function g∈Cs[0,1] are unknown. We will prove error bounds for both algorithms and provide an extensive numerical comparison of both. We further propose an approach of how to apply these algorithms to capture general sleeve functions, which are constant along some lower dimensional submanifolds.

Atiyah–Patodi–Singer η $$ \eta $$ -Invariant and Invariants of Finite Degree

Let η be the Atiyah–Patodi–Singer invariant considered on smooth, compact, oriented, threedimensional submanifolds of ℝ n , and let A be an additive subgroup of ℝ. The problem of computing the degree of invariants of the form η mod A is examined. Here, the functional definition of invariants of finite degree is used. (A similar approach is used in S. S. Podkorytov’s paper “Quadratic property of the rational semicharacteristic.”) The main results are as follows. If 1 ∉ A, then the degree is infinite. If $$ \frac{1}{3}\in A $$ , then the degree is equal to 1. Bibliography: 10 titles.

Higher codimensional Euclidean helix submanifolds

A submanifold of Rn whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find every solution to the eikonal PDE on a Riemannian manifold locally. As a consequence we give a local construction of arbitrary Euclidean helix submanifolds of any dimension and codimension. Also we characterize the ruled helix submanifolds and in particular we describe those which are minimal.

Manifold Homotopy via the Flow Complex

AbstractIt is known that the critical points of the distance function induced by a dense sample P of a submanifold Σ of ℝn are distributed into two groups, one lying close to Σ itself, called the shallow, and the other close to medial axis of Σ, called deep critical points. We prove that under (uniform) sampling assumption, the union of stable manifolds of the shallow critical points have the same homotopy type as Σ itself and the union of the stable manifolds of the deep critical points have the homotopy type of the complement of Σ. The separation of critical points under uniform sampling entails a separation in terms of distance of critical points to the sample. This means that if a given sample is dense enough with respect to two or more submanifolds of ℝn, the homotopy types of all such submanifolds together with those of their complements are captured as unions of stable manifolds of shallow versus those of deep critical points, in a filtration of the flow complex based on the distance of critical points to the sample. This results in an algorithm for homotopic manifold reconstruction when the target dimension is unknown.

Legendrian contact homology in 𝑃×ℝ

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × R P\times \mathbb {R} , where P P is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of R n \mathbb {R}^n and, more generally, invariants of self transverse immersions into R n \mathbb {R}^n up to restricted regular homotopies. When n = 3 n=3 , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

Isoparametric Riemannian Submanifolds of ℝn,k

We prove that, essentially, the only complete, irreducible, and isoparametric Riemannian submanifolds of ℝn,k are hyperbolic spaces or isoparametric submanifolds of Euclidean spaces. In particular, a simple and geometric proof of the theorem of Wu is derived.

Focal Loci of Algebraic Hypersurfaces: A General Theory

The focal locus is traditionally defined for a differentiable submanifold of Rn. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space \(P_C^n \), once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in \(P_C^n \). Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.

Submanifolds and their k-planar number

IfM is a submanifold of ℝn the normal plane together with a tangent at a pointx∈ M span an affine plane which intersectsM, locally, in a curve called the normal section. This paper introduces, for each integerk, a parameter calledπ(k) which depends on the curvature and torsion of these normal sections. Results are obtained showing how these parameters for a product submanifold depend on the value of this parameter for the factor submanifolds.

Partial tubes about immersed manifolds

We consider a generalization of the idea of an e-tube about a submanifold of ℝn which includes, on the one hand, submanifolds parallel to the original and, on the other, isoparametric submanifolds about a focal submanifold. We discuss properties that are inherited from the core manifold and the type fibre. The construction is used to show that there are submanifolds with ‘many different’ parallel submanifolds.

Infinitesimally homogeneous submanifolds of euclidean spaces

For a class of submanifolds of ℝN, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativess≤k+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ℝ N , whose orbit atp is a complete submanifold that extendsM.

Transcendental submanifolds of Rn

In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set is related to the real algebraic cycles of the real part. As a corollary we give examples of smooth submanifolds of a Euclidean space which can not be isotoped to real parts of complex nonsingular subvarieties in the corresponding projective space.

On the p-Thin Problem for Hypersurfaces of Rn With Zero Gaussian Curvature

AbstractA subset M of Rn is said to be p-thin if T ∊ FLP(Rn) and supp(T) ⊂ M imply T = 0. For a class of smooth (n — 1 )-dimensional submanifolds of Rn, we obtain the optimal result for the p-thin problem, which is applied to give the complete solution to a uniqueness problem of wave equations.

The Gauss map for Kählerian submanifolds of 𝑅ⁿ

We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.

Complex cycles on real algebraic models of a smooth manifold

Let M M be a compact connected orientable C ∞ {C^\infty } submanifold of R n {\mathbb {R}^n} with 2 dim ⁡ M + 1 ≤ n 2\dim M + 1 \leq n . Let G G be a subgroup of H 2 ( M , Z ) {H^2}(M,\mathbb {Z}) such that the quotient group H 2 ( M , Z ) {H^2}(M,\mathbb {Z}) has no torsion. Then M M can be approximated in R n {\mathbb {R}^n} by a nonsingular algebraic subset X X such that H C - a l g 2 ( X , Z ) H_{\mathbb {C} \operatorname {- alg}}^{2}(X,\mathbb {Z}) is isomorphic to G G . Here H C - a l g 2 ( X , Z ) H_{\mathbb {C}\operatorname { - alg}}^2(X,\mathbb {Z}) denotes the subgroup of H 2 ( X , Z ) {H^2}(X,\mathbb {Z}) generated by the cohomology classes determined by the complex algebraic hypersurfaces in a complexification of X X .

Boundary regularity for solutions of the equation of prescribed Gauss curvature

We study the boundary regularity of convex solutions of the equation of prescribed Gauss curvature in a domain Ω ⊂ ℝn in the case that the gradient of the solution is infinite on some relatively open, uniformly convex portion Γ of ∂Ω. Under suitable conditions on the data we show that near Γ × ℝ the graph of u is a smooth hypersurface (as a submanifold of ℝn + 1) and that u|Γ is smooth. In particular, u is Hölder continuous with exponent 1/2 near Г.

A network model for incompressible two‐fluid flow and its numerical solution

AbstractIn this paper we consider an incompressible version of the two‐fluid network model proposed by Porsching (Nu. Methods Part. Diff. Eq., 1, 295–313 [1985]). The system of equations governing the model is a mixed system of differential and algebraic equations (DAEs). These DAEs are then recast, through proper transformation, into a system of ordinary differential equations on a submanifold of ℝn, for which uniqueness, existence, and stability theorems are proved. Numerical simulations are presented.

Compact conformally flat hypersurfaces

Roughly speaking, a conformal space is a differentiable manifold M n {M^n} in which the notion of angle of tangent vectors at a point p ∈ M n p \in {M^n} makes sense and varies differentiably with p p ; two such spaces are (locally) equivalent if they are related by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space R n {R^n} . A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure; in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, assume the ambient space to be R n + 1 {R^{n + 1}} . Then, if n ⩾ 4 n \geqslant 4 , a conformally flat hypersurface M n ⊂ R n + 1 {M^n} \subset {R^{n + 1}} can be described as follows. Diffeomorphically, M n {M^n} is a sphere S n {S^n} with b 1 ( M ) {b_1}(M) handles attached, where b 1 ( M ) {b_1}(M) is the first Betti number of M M . Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of R n + 1 {R^{n + 1}} that are foliated by complete round ( n − 1 ) (n - 1) -spheres and are joined through their boundaries to the following three types of umbilic submanifolds of R n + 1 {R^{n + 1}} : (a) an open piece of an n n -sphere or an n n -plane bounded by round ( n − 1 ) (n - 1) -sphere, (b) a round ( n − 1 ) (n - 1) -sphere, (c) a point.