Dimensional Submanifolds Research Articles

A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach

In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a connection to SPDEs in continuously embedded spaces.

Non-relativistic expansion of open strings and D-branes

We expand the relativistic open bosonic string in powers of 1/c2 where c is the speed of light. We perform this expansion to next-to-leading order in 1/c2 and relate our results to known descriptions of non-relativistic open strings obtained by taking limits. Just as for closed strings the non-relativistic expansion is well-defined if the open string winds a circle in the target space. This direction must satisfy Dirichlet boundary conditions. It is shown that the endpoints of the open string behave as Bargmann particles in the non-relativistic regime. These open strings end on nrDp-branes with p ≤ 24. When these nrDp-branes do not fluctuate they correspond to (p + 1)-dimensional Newton-Cartan submanifolds of the target space. When we include fluctuations and worldvolume gauge fields their dynamics is described by a non-relativistic version of the DBI action whose form we derive from symmetry considerations. The worldvolume gauge field and scalar field of a nrD24-brane make up the field content of Galilean electrodynamics (GED), and the effective theory on the nrD24-brane is precisely a non-linear version of GED. We generalise these results to actions for any nrDp-brane by demanding that they have the same target space gauge symmetries that the non-relativistic open and closed string actions have. Finally, we show that the nrDp-brane action is transverse T-duality covariant. Our results agree with the findings of Gomis, Yan and Yu in [1].

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

On the first eigenvalue of the Laplacian for polygons

A 2006 conjecture of Antunes and Freitas is addressed connecting the scaling-invariant polygonal isoperimetric and principal frequency deficits for triangles. This yields a quantitative polygonal Faber–Krahn inequality for triangles with an explicit constant. Furthermore, a problem mentioned in the 1951 book “Isoperimetric Inequalities In Mathematical Physics” by Pólya and Szegö is addressed: a formula is given for the principal frequency of a triangle. Moreover, a space of polygons is constructed for the classical Pólya and Szegö problem: in 1947, Pólya proved that if n = 3, 4 the regular polygon Pn minimizes the principal frequency of an n-gon with given area α > 0 and suggested that the same holds when n ≥ 5. In 1951, Pólya and Szegö discussed the possibility of counterexamples. This paper constructs explicit (2n − 4)–dimensional polygonal manifolds M(n,α) and proves the existence of a computable N ≥ 5 such that for all n ≥ N, the admissible n-gons are given via M(n,α)and there exists an explicit set An(α)⊂M(n,α) such that Pn has the smallest principal frequency among n-gons in An(α). Inter-alia when n ≥ 3, a formula is proved for the principal frequency of a convex P∈M(n,α)in terms of an equilateral n-gon with the same area; and, the set of equilateral polygons is proved to be an (n − 3)–dimensional submanifold of the (2n − 4)–dimensional manifold M(n,α)near Pn. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and W2,p/BMO estimates. Last, an application is given in the context of electron bubbles.

Some five dimensional persistent submanifolds of the Study quadric

This work studies persistent submanifolds of the group of rigid-body displacements. At each point of a submanifold the tangent space, translated to the identity in the group becomes a screw system. For persistent submanifolds these screws systems are mutually conjugate by a rigid-body transformation. We present novel results on persistent submanifolds composed of three subgroups and products of subgroups with persistent submanifolds. We also show that some submanifolds defined as solutions to geometric constraint problems are persistent.Examples of 5-dimensional persistent submanifolds are then studied. We show that many examples can be show to be persistent in several ways. We also show that there are large overlaps between different types of submanifolds. Many constraint varieties can be realised by open-loop kinematic chains and many of these persistent 5-dimensional submanifolds are the intersection of the Study quadric with algebraic hypersurfaces. Many of the examples considered are well known but we also introduce several novel examples of 5-dimensional persistent submanifolds.

Carrollian structure of the null boundary solution space

We study pure D dimensional Einstein gravity in spacetimes with a generic null boundary. We focus on the symplectic form of the solution phase space which comprises a 2D dimensional boundary part and a 2(D(D − 3)/2 + 1) dimensional bulk part. The symplectic form is the sum of the bulk and boundary parts, obtained through integration over a codimension 1 surface (null boundary) and a codimension 2 spatial section of it, respectively. Notably, while the total symplectic form is a closed 2-form over the solution phase space, neither the boundary nor the bulk symplectic forms are closed due to the symplectic flux of the bulk modes passing through the boundary. Furthermore, we demonstrate that the D(D − 3)/2 + 1 dimensional Lagrangian submanifold of the bulk part of the solution phase space has a Carrollian structure, with the metric on the D(D − 3)/2 dimensional part being the Wheeler-DeWitt metric, and the Carrollian kernel vector corresponding to the outgoing Robinson-Trautman gravitational wave solution.

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

<abstract><p>In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.</p></abstract>

On the Robustness of Bayesian Neural Networks to Adversarial Attacks.

Vulnerability to adversarial attacks is one of the principal hurdles to the adoption of deep learning in safety-critical applications. Despite significant efforts, both practical and theoretical, training deep learning models robust to adversarial attacks is still an open problem. In this article, we analyse the geometry of adversarial attacks in the over-parameterized limit for Bayesian neural networks (BNNs). We show that, in the limit, vulnerability to gradient-based attacks arises as a result of degeneracy in the data distribution, i.e., when the data lie on a lower dimensional submanifold of the ambient space. As a direct consequence, we demonstrate that in this limit, BNN posteriors are robust to gradient-based adversarial attacks. Crucially, by relying on the convergence of infinitely-wide BNNs to Gaussian processes (GPs), we prove that, under certain relatively mild assumptions, the expected gradient of the loss with respect to the BNN posterior distribution is vanishing, even when each NN sampled from the BNN posterior does not have vanishing gradients. The experimental results on the MNIST, Fashion MNIST, and a synthetic dataset with BNNs trained with Hamiltonian Monte Carlo and variational inference support this line of arguments, empirically showing that BNNs can display both high accuracy on clean data and robustness to both gradient-based and gradient-free adversarial attacks.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Bounds for Eigenvalues of q-Laplacian on Contact Submanifolds of Sasakian Space Forms

This study establishes new upper bounds for the mean curvature and constant sectional curvature on Riemannian manifolds for the first positive eigenvalue of the q-Laplacian. In particular, various estimates are provided for the first eigenvalue of the q-Laplace operator on closed orientated (l+1)-dimensional special contact slant submanifolds in a Sasakian space form, M˜2k+1(ϵ), with a constant ψ1-sectional curvature, ϵ. From our main results, we recovered the Reilly-type inequalities, which were proven before this study.

Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian(Pλ){(−Δ)su=λu−q+u2s⁎−1,u>0in Ω,A(u)=0on∂Ω=∑D∪∑N, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, 1/2<s<1, λ>0 is a real parameter, 0<q<1, N>2s, 2s⁎=2N/(N−2s) andA(u)=uX∑D+∂νuX∑N,∂ν=∂∂ν. Here ∑D, ∑N are smooth (N−1) dimensional submanifolds of ∂Ω such that ∑D∪∑N=∂Ω, ∑D∩∑N=∅ and ∑D∩∑N‾=τ′ is a smooth (N−2) dimensional submanifold of ∂Ω. Within a suitable range of λ, we establish existence of at least two opposite energy solutions for (Pλ) using the standard Nehari manifold technique.

On periodically modulated rolls in the generalized Swift–Hohenberg equation: Galerkin’ approximations

We study in this paper the existence of periodically modulated in one variable and localized in another variable solutions to the cubic Swift–Hohenberg equation on the plane R2. In the first part we try to apply the method by Kirschgässner–Mielke to reduce the problem to the search of finite dimensional submanifolds with periodic orbits on them in some formal infinite-dimensional dynamical system generated by the stationary SH equation. It turns out that it cannot be done immediately due to properties of the spectrum for the linearized system at the localized roll (a one-dimensional pulse). In the second part we change roles of variables and formulate the problem as funding homoclinic orbits to an equilibrium of the formal infinite-dimensional system in the space of periodic functions in variable y. Staying apart the proof of the exact theorem on the existence of related center manifold, we exploit the Bubnov–Galerkin method to derive the Hamiltonian finite-dimensional ODEs with four or six degrees of freedom having the saddle type equilibrium whose homoclinic orbits correspond to approximate solution on needed type. The search for homoclinic orbits is performed by means of numerical methods.

Some Geometric Inequalities by the ABP Method

Abstract In this paper, we apply the so-called Alexandrov–Bakelman–Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $\Sigma $ of $\mathbb S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb S^{n}| \leq |\Sigma |$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for positive symmetric two-tensors on smooth domains in $\mathbb R^{n}$, which was established by D. Serre when the domain is convex. In the last application of the ABP method, we formulate and prove an inequality related to quermassintegrals of closed hypersurfaces of the Euclidean space.

Curved thin-film limits of chiral Dirichlet energies

We investigate the curved thin-film limit of a family of perturbed Dirichlet energies in the space of H1 Sobolev maps defined in a tubular neighborhood of an (n−1)-dimensional submanifold N of Rn and with values in an (m−1)-dimensional submanifold M of Rm. The perturbation K that we consider is represented by a matrix-valued function defined on M and with values in Rm×n. Under natural regularity hypotheses on N, M, and K, we show that the family of these energies converges, in the sense of Γ-convergence, to an energy functional on N of an unexpected form, which is of particular interest in the theory of magnetic skyrmions. As a byproduct of our results, we get that in the curved thin-film limit, antisymmetric exchange interactions also manifest under an anisotropic term whose specific shape depends both on the curvature of the thin film and the curvature of the target manifold. Various types of antisymmetric exchange interactions in the variational theory of micromagnetism are a source of inspiration and motivation for our work.

On gap rigidity problems for compact Hermitian symmetric spaces

We prove a gap rigidity theorem for diagonal curves in maximal polyspheres of irreducible compact Hermitian symmetric spaces of tube type, which is a dual analog to a theorem obtained by Mok. Motivated by the proof we show that the Chow space of certain totally geodesic submanifolds is affine algebraic, which gives a weaker version of gap rigidity for a class of higher dimensional submanifolds.

Connected sums of codimension two locally flat submanifolds

Let $X$ and $Y$ be oriented topological manifolds of dimension $n\!+\!2$ , and let $K\! \subset \! X$ and $J \! \subset \! Y$ be connected, locally-flat, oriented, $n$ –dimensional submanifolds. We show that up to orientation preserving homeomorphism there is a well-defined connected sum $(X,K)\! \mathbin {\#}\! (Y,J)$ . For $n = 1$ , the proof is classical, relying on results of Rado and Moise. For dimensions $n=3$ and $n \ge 6$ , results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For $n = 2, 4,$ and $5$ , Freedman and Quinn's work on topological four-manifolds is required along with the higher dimensional theory.

Transport Type Metrics on the Space of Probability Measures Involving Singular Base Measures

We develop the theory of a metric, which we call the $$\nu $$ -based Wasserstein metric and denote by $$W_\nu $$ , on the set of probability measures $${\mathcal {P}}(X)$$ on a domain $$X \subseteq \mathbb {R}^m$$ . This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $$\nu $$ and is relevant in particular for the case when $$\nu $$ is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $$\nu $$ -based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $$\nu $$ ; we also characterize it in terms of integrations of classical Wasserstein metric between the conditional probabilities when measures are disintegrated with respect to optimal transport to $$\nu $$ , and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $$W_\nu $$ , defined relative to a fixed based measure $$\mu $$ , on the set of measures which are absolutely continuous with respect to a second fixed based measure $$\sigma $$ . As we vary the base measure $$\nu $$ , the $$\nu $$ -based Wasserstein metric interpolates between the usual quadratic Wasserstein metric (obtained when $$\nu $$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $$\nu $$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When $$\nu $$ concentrates on a lower dimensional submanifold of $$\mathbb {R}^m$$ , we prove that the variational problem in the definition of the $$\nu $$ -based Wasserstein metric has a unique solution. We also establish geodesic convexity of the usual class of functionals, and of the set of source measures $$\mu $$ such that optimal transport between $$\mu $$ and $$\nu $$ satisfies a strengthening of the generalized nestedness condition introduced in McCann and Pass (Arch Ration Mech Anal 238(3):1475–1520, 2020). We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.

Gap type results for spacelike submanifolds with parallel mean curvature vector

We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, , which has a non-trivial representation as a 2N − 2 dimensional submanifold of (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.

A Canonical Neighborhood Theorem for Mean Curvature Flow in Higher Codimension

Abstract In dimensions $n \geq 5$, we prove a canonical neighborhood theorem for the mean curvature flow of compact $n$-dimensional submanifolds in $\mathbb {R}^N$ satisfying a pinching condition $|A|^2 &amp;lt; c|H|^2$ for $c = \min \{ \frac {3(n+1)}{2n(n+2)},\frac {1}{n-2}\}.$