Critical Submanifolds Research Articles

The Morse Index for Manifolds with Constant Sectional Curvature

We compute the Morse index of a critical submanifold of the energy functional on the loop space of a manifold with constant sectional curvature. The case of constant non-positive sectional curvature is a known result and the case of a sphere has been proved by Klingenberg. We adapt Klingenberg’s proof of the case of a sphere to the case of constant sectional curvature, to obtain the possible Morse indices of critical submanifolds of the energy functional.

Homotopic Distance and Generalized Motion Planning

We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifolds of any Morse–Bott function. This generalizes the Lusternik–Schnirelmann theorem (for Morse functions) and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

Information geometry of quantum critical submanifolds: Relevant, marginal, and irrelevant operators

We analyze the thermodynamical limit of the quantum metric along critical submanifolds of theory space. Building upon various results previously known in the literature, we relate its singular behavior to normal directions, which are naturally associated with relevant operators in the renormalization group sense. We formulate these results in the language of information theory and differential geometry. We exemplify our theory through the paradigmatic examples of the XY and Haldane models, where the normal directions to the critical submanifolds are seen to be precisely those along which the metric has singular behavior, while for the tangent ones it vanishes -- these directions lie in the kernel of the metric.

Embedding Principle: A Hierarchical Structure of Loss Landscape of Deep Neural Networks

We prove a general Embedding Principle of loss landscape of deep neural networks (NNs) that unravels a hierarchical structure of the loss landscape of NNs, i.e., loss landscape of an NN contains all critical points of all the narrower NNs. This result is obtained by constructing a class of critical embeddings which map any critical point of a narrower NN to a critical point of the target NN with the same output function. By discovering a wide class of general compatible critical embeddings, we provide a gross estimate of the dimension of critical submanifolds embedded from critical points of narrower NNs. We further prove an irreversiblility property of any critical embedding that the number of negative/zero/positive eigenvalues of the Hessian matrix of a critical point may increase but never decrease as an NN becomes wider through the embedding. Using a special realization of general compatible critical embedding, we prove a stringent necessary condition for being a "truly-bad" critical point that never becomes a strict-saddle point through any critical embedding. This result implies the commonplace of strict-saddle points in wide NNs, which may be an important reason underlying the easy optimization of wide NNs widely observed in practice.

Sharp estimates for the first eigenvalue of Schrödinger operator in the unit sphere

Let H \mathbf {H} be the mean curvature vector of an n n -dimensional submanifold M n M^n in a Riemannan manifold. The critical submanifolds of the total mean curvature functional H = ∫ | H | n \mathcal H=\int |\mathbf {H}|^n are called H \mathcal H -submanifolds. In this note, we will prove that compact Legendrian H \mathcal H -surfaces in the unit sphere S 5 \mathbb {S}^{5} are minimal. We also investigate the first eigenvalue of the Schrödinger operator L = − Δ − q L=-\Delta -q on M 2 M^2 and M 3 M^3 , where q q is some potential function, and obtain a sharp estimate for the first eigenvalue of L L .

Variations of hodge structures of rank three k-Higgs bundles and moduli spaces of holomorphic triples

There is an isomorphism between the moduli spaces of \(\sigma \)-stable holomorphic triples and some of the critical submanifolds of the moduli space of k-Higgs bundles of rank three, whose elements \((E,\varphi ^k)\) correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of k-Higgs bundles of rank three. The main objective here is to study the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.

Non-linear Morse–Bott functions on quaternionic Stiefel manifolds

In the Stiefel manifold Xn,k, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n−2k. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

Dynamical system analysis of brane induced gravity with tachyon field

In this manuscript we use the dynamical system approach to study the linear dynamics of a normal DGP brane-world model with a tachyon field as the dark energy component. Our focus is on a Gaussian tachyonic potential in which the parameter , goes to infinity. One of the most important results of this study is that we find critical submanifolds which indicate the effect of extra dimension.

Height functions on compact symmetric spaces

We consider height functions on symmetric spaces \(M\cong G/K\) embedded in the associated matrix Lie group \(G\). In particular we study the relationship between the critical sets of the height function on \(G\) and its restriction to \(M\). Also we prove that the gradient flow on \(M\) can be integrated by means of a generalized Cayley transform; this allows to obtain explicit local charts for the critical submanifolds. Finally, we discuss how to reduce the study of any height function to the case where the ground hyperplane is orthogonal to a real diagonal matrix. This result requires to prove the existence of a polar decomposition adapted to the automorphism defining \(M\). Detailed examples are given.

Cascades and perturbed Morse–Bott functions

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{FraTheA}. Similar data, which also includes a parameter $\epsilon > 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_\epsilon:M \rightarrow \mathbb{R}$ \cite{AusMor} \cite{BanDyn}. In this paper we show that the Morse-Smale-Witten chain complex of $h_\epsilon$ is the same as the Morse chain complex defined using cascades for any $\epsilon >0$ sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of $f:M \rightarrow \mathbb{R}$ is isomorphic to the singular homology $H_\ast(M;\mathbb{Z})$.

Parallel calibrations and minimal submanifolds

Given a parallel calibration $\varphi\in\Omega^{p}(M)$ on a Riemannian manifold $M$, I prove that the $\varphi$-critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\varphi$-critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^{\infty}(M)$-linear subspace $\mathscr{P}\subset\Omega^{p}(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi–Yau manifolds, (co)associative calibrations on $G_{2}$-manifolds, and the Cayley calibration on $\operatorname{Spin}(7)$-manifolds.)

The Morse–Bott inequalities via a dynamical systems approach

AbstractLet f:M→ℝ be a Morse–Bott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré polynomial of M. We prove that R(t) is a polynomial with non-negative integer coefficients by showing that the number of gradient flow lines of the perturbation of f between two critical points p,q∈Cj of relative index one coincides with the number of gradient flow lines between p and q of the Morse function fj. This leads to a relationship between the kernels of the Morse–Smale–Witten boundary operators associated to the Morse functions fj and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients are used.

Optimal control landscape for the generation of unitary transformations

The generation of specific unitary transformations is central to a variety of quantum control problems. Given a target unitary transformation, the optimal control landscape is defined as the Hilbert-Schmidt distance between the target and controlled unitary transformation as a function of the control variables. The critical topology of the landscape is analyzed for controllable quantum systems evolving under unitary dynamics over a finite dimensional Hilbert space. It is found that the critical regions of the landscape corresponding to global optima are isolated points, and the local optima are Grassmannian submanifolds. The volumes of the critical submanifolds corresponding to suboptimal critical values asymptotically vanish in the limit of large Hilbert space dimension. Furthermore, these critical submanifolds have saddle-point topology, which cannot act as traps when searching for optimal controls. These favorable properties of the local optima suggest that the landscape topology is generally amenable to optimization. The analysis is independent of the particular structure of the system Hamiltonian, except for the assumption of full controllability, and the results are universal to the control of unitary transformations of any quantum system.

Control landscapes for two-level open quantum systems

A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper, the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive trace-preserving maps (i.e., Kraus maps), are investigated in detail. The objective function, which is the expectation value of a target system operator, is defined on the Stiefel manifold representing the space of Kraus maps. Three practically important properties of the objective function are found: (a) the absence of local maxima or minima (i.e., false traps), (b) the existence of multi-dimensional sub-manifolds of optimal solutions corresponding to the global maximum and minimum and (c) the connectivity of each level set. All of the critical values and their associated critical sub-manifolds are explicitly found for any initial system state. Away from the absolute extrema there are no local maxima or minima, and only saddles may exist, whose number and the explicit structure of the corresponding critical sub-manifolds are determined by the initial system state. There are no saddles for pure initial states, one saddle for a completely mixed initial state, and two saddles for partially mixed initial states. In general, the landscape analysis of critical points and optimal manifolds is relevant to explain the relative ease of obtaining good optimal control outcomes in the laboratory, even in the presence of the environment.

Characterization of the critical submanifolds in quantum ensemble control landscapes

The quantum control landscape is defined as the functional that maps the control variables to the expectation values of an observable over the ensemble of quantum systems. Analyzing the topology of such landscapes is important for understanding the origins of the increasing number of laboratory successes in the optimal control of quantum processes. This paper proposes a simple scheme to compute the characteristics of the critical topology of the quantum ensemble control landscapes showing that the set of disjoint critical submanifolds one-to-one corresponds to a finite number of contingency tables that solely depend on the degeneracy structure of the eigenvalues of the initial system density matrix and the observable whose expectation value is to be maximized. The landscape characteristics can be calculated as functions of the table entries, including the dimensions and the numbers of positive and negative eigenvalues of the Hessian quadratic form of each of the connected components of the critical submanifolds. Typical examples are given to illustrate the effectiveness of this method.

Lagrangian skeletons, periodic geodesic flows and symplectic cuttings

In this paper, we investigate symplectic manifolds endowed with a Morse–Bott function with only two critical submanifolds, one of which is Lagrangian while the other one is symplectic.

A variational problem for submanifolds in a sphere

Let $ x{:}\ M\rightarrow S^{n+p} $ be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S n + p , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional $\int_M(S-nH^2)^{\frac{n}{2}}d\nu$ , where $ S=\sum_{\alpha,i,\,j}(h^\alpha_{ij})^2 $ is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S n + p . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional $\int_M(S-nH^2)d\nu$ . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.

Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.

Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds

The purpose of this article is to study conformal harmonic maps f: Σ → M, where Σ is a closed Riemann surface and M is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold M is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of E. They are Morse nondegenerate in the usual sense if Σ has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if S has genus one, and on six-dimensional nondegenerate critical submanifolds if Σ has genus zero.

TOPOLOGY OF $U(2, 1)$ REPRESENTATION SPACES

The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2, 1) and SU(2, 1) are calculated. The results are obtained using the identification of these moduli spaces with moduli spaces of Higgs bundles, and Morse theory, following Hitchin's programme. This requires a careful analysis of critical submanifolds, which turn out to have a description using either symmetric products of the surface or moduli spaces of Bradlow pairs. 2000 Mathematics Subject Classification 14H60 (primary); 14H30, 32C18 (secondary).