Target Manifold Research Articles

A DDVV Conjecture for Riemannian Maps

The Wintgen inequality is a significant result in the field of differential geometry, specifically related to the study of submanifolds in Riemannian manifolds. It was discovered by Pierre Wintgen. In the present work, we deal with the Riemannian maps between Riemannian manifolds that serve as a superb method for comparing the geometric structures of the source and target manifolds. This article is the first to explore a well-known conjecture, called DDVV inequality (a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken), for Riemannian maps, where we consider different space forms as target manifolds. There are numerous research problems related to such inequality in various ambient manifolds. These problems can all be explored within the general framework of Riemannian maps between various Riemannian manifolds equipped with notable geometric structures.

Liouville type theorems for generalized P-harmonic maps

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature.

Łojasiewicz inequalities for almost harmonic maps near simple bubble trees

We prove Łojasiewicz inequalities for the harmonic map energy for maps from surfaces of positive genus into general analytic target manifolds which are close to simple bubble trees and as a consequence obtain new results on the convergence of harmonic map flow and on the energy spectrum of harmonic maps with small energy. Our results and techniques are not restricted to particular targets or to integrable settings and we are able to lift general Łojasiewicz–Simon inequalities valid near harmonic maps \hat{\omega}\colon S^{2}\to N to the singular setting whenever the bubble \hat{\omega} is attached at a point which is not a branch point.

Lie-algebraic Kähler sigma models with U(1) isotropy

We discuss various questions that emerge in connection with the Lie-algebraic deformation of the CP1 sigma model in two dimensions. First, we supersymmetrize the original model endowing it with the minimal N=(0,2) and extended N=(2,2) supersymmetries. Then we derive the general hypercurrent anomaly in both cases. In the latter case this anomaly is one-loop but is somewhat different from the standard expressions one can find in the literature because the target manifold is nonsymmetric. We also show how to introduce the twisted masses and the θ term, and study the Bogomol’nyi–Prasad–Sommerfield equation for instantons, in particular the value of the topological charge. Then we demonstrate that the second loop in the β function of the Lie-algebraic sigma model is due to an infrared effect. To this end we use a regularization. We also conjecture that the above statement is valid for higher loops too, similar to the parallel phenomenon in four-dimensional N=1 super-Yang-Mills. In the second part of the paper we develop a special dimensional reduction—namely, starting from the two-dimensional Lie-algebraic model we arrive at a quasi-exactly solvable quantum-mechanical problem of the Lamé type. Published by the American Physical Society 2024

On stability of subelliptic harmonic maps with potential

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension ≥3.

Smooth generalized symmetries of quantum field theories

Dynamical quantum field theories (QFTs), such as those in which spacetimes are equipped with a metric and/or a field in the form of a smooth map to a target manifold, can be formulated axiomatically using the language of ∞-categories. According to a geometric version of the cobordism hypothesis, such QFTs collectively assemble themselves into objects in an ∞-topos of smooth spaces. We show how this allows one to define and study generalized global symmetries of such QFTs. The symmetries are themselves smooth, so the ‘higher-form’ symmetry groups can be endowed with, e.g., a Lie group structure.Among the more surprising general implications for physics are, firstly, that QFTs in spacetime dimension d, considered collectively, can have d-form symmetries, going beyond the known (d−1)-form symmetries of individual QFTs and, secondly, that a global symmetry of a QFT can be anomalous even before we try to gauge it, due to a failure to respect either smoothness (in that a symmetry of an individual QFT does not smoothly extend to QFTs collectively) or locality (in that a symmetry of an unextended QFT does not extend to an extended one).Smoothness anomalies are shown to occur even in 2-state systems in quantum mechanics (here formulated axiomatically by equipping d=1 spacetimes with a metric, an orientation, and perhaps some unitarity structure). Locality anomalies are shown to occur even for invertible QFTs defined on d=1 spacetimes equipped with an orientation and a smooth map to a target manifold. These correspond in physics to topological actions for a particle moving on the target and the relation to an earlier classification of such actions using invariant differential cohomology is elucidated.

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}. For each d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 4$$\\end{document}, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

We show how the tangent functor extends from ordinary smooth maps to “microformal morphisms” (also called “thick morphisms”) of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as “non-linear chain maps” that can induce non-linear transformations of cohomology.

Stability of isometric immersions of hypersurfaces

Abstract We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Automated mixing of maximally localized Wannier functions into target manifolds

Maximally localized Wannier functions (MLWFs) are widely used in electronic-structure calculations. We have recently developed automated approaches to generate MLWFs that represent natural tight-binding sets of atomic-like orbitals; these describe accurately both the occupied states and the complementary unoccupied ones. For many applications, it is required to use MLWFs that describe instead certain target groups of bands: the valence or the conduction bands, or correlated manifolds. Here, we start from these tight-binding sets of MLWFs, and mix them using a combination of parallel transport and maximal localization to construct manifold-remixed Wannier functions (MRWFs): these are orthogonal sets of MLWFs that fully and only span desired target submanifolds. The algorithm is simple and robust, and is showcased here in reference applications (silicon, MoS2, and SrVO3) and in a mid-throughput study of 77 insulators.

Application of a Stochastic Extension of the Analytical Design of Aggregated Regulators to a Multidimensional Biomedical Object

The results of the application of the methods of the synergetic control theory to a high-dimensional immunology object with uncertainty in its descriptions are reported. The control here is the therapy treated as a problem for constructing an optimal cure program. The control object is presented in continuous and discrete forms, i.e., mathematical models given by a system of ordinary differential equations with a bounded disturbance and a system of stochastic difference equations, respectively. Two algorithms for deriving robust regulators applicable to a 10-dimensional nonlinear multi-loop system with unstable limit states, which models an immune response to the hepatitis B infection, are obtained. Analytical control design for a continuous model relies on the method of nonlinear adaptation on the target manifold. The second algorithm represents a stochastic extension of the method of analytical design of aggregated discrete regulators minimizing the variance of the target macro variable. The numerical simulation of the developed control systems indicates the performance of the designed control algorithms. The results of this study can be used as a component part of the mathematical tools of expert systems and decision support systems.

η-Ricci–Yamabe Solitons along Riemannian Submersions

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the η-Ricci–Yamabe soliton (η-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an η-RY soliton, an η-Ricci soliton, and an η-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an η-RY soliton, an η-Ricci soliton, an η-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field ω of the soliton is of gradient type =:grad(γ) and provide some examples of an η-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.

CP2 skyrmions and skyrmion crystals in realistic quantum magnets

Magnetic skyrmions are nanoscale topological textures that have been recently observed in different families of quantum magnets. These objects are called CP1 skyrmions because they are built from dipoles—the target manifold is the 1D complex projective space, CP1 ≅ S2. Here we report the emergence of magnetic CP2 skyrmions in a realistic spin-1 model, which includes both dipole and quadrupole moments. Unlike CP1 skyrmions, CP2 skyrmions can also arise as metastable textures of quantum paramagnets, opening a new road to discover emergent topological solitons in non-magnetic materials. The quantum phase diagram of the spin-1 model also includes magnetic field-induced CP2 skyrmion crystals that can be detected with regular momentum- (diffraction) and real-space (Lorentz transmission electron microscopy) experimental techniques.

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.

Truncated Affine Rozansky–Witten Models as Extended TQFTs

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T^*mathbb {C}^n. The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category mathcal C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in mathcal C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them.

Top-Down Holography in an Asymptotically Flat Spacetime

We propose a holographic duality for a four dimensional Wess-Zumino-Witten model with target manifold SO(8), coupled to scalar-flat Kähler gravity on an asymptotically flat, four dimensional background known as the Burns metric. The holographic dual is a two dimensional chiral algebra built out of gauged β-γ systems with SO(8) flavor. We test the duality by matching two-point correlators of soft gluon currents with two-point gluon amplitudes, and their leading operator product expansion coefficients with collinear limits of three-point gluon amplitudes.

Deep Domain Adaptation Using Cascaded Learning Networks and Metric Learning

How can we bridge efficiently distribution difference between the source and target domains in an isomorphic latent feature space by using metric learning. This study introduces metric learning on manifolds which combine a cascaded learning network and a metric learning model to form a Unified Domain Adaptation Model. Our approach is based on formulating a transfer from the source to the target as a geometric mean metric learning problem on manifolds. The solution of symmetric positive definite covariance matrices not only reduces the statistical differences between the source and target domains, but also the underlying geometry of the source and target domains using diffusions on the underlying source and target manifolds. By retaining both the nonlinear structure of the Riemannian geometry of the open cone of symmetric positive definite matrices and cascaded learning networks, we improve the state-of-the-art results on the Amazon (A), Caltech256 (C), DSLR (D), Webcam (W) and VisDA benchmark datasets by knowledge transfer, while achieving comparable performances to competing methods on domain adaptation modeling.

Half-harmonic gradient flow: aspects of a non-local geometric PDE

<abstract><p>The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in <sup>[<xref ref-type="bibr" rid="b47">47</xref>]</sup> is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see <sup>[<xref ref-type="bibr" rid="b40">40</xref>]</sup>, is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $ t \to +\infty $, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $ S^{n-1} $, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $ N $ (cf. <sup>[<xref ref-type="bibr" rid="b48">48</xref>]</sup>).</p></abstract>

Passivity and Immersion (P&I) Approach for Constructive Stabilization and Control of Nonlinear Systems

This letter proposes a control algorithm as a stepping stone for the stabilization and control of structured and unstructured systems discussed in the literature. The foundation behind this proposed controller design theory utilizes an invariant target manifold giving rise to a non-degenerate form, through which the definition of certain passive outputs and storage functions provide the control law for stabilizing the system. The proposed control framework is developed by integrating the notions of immersion and passivity hence labeled as the “Passivity and Immersion (P&I) approach.” Immersing the system dynamics into the user-defined lower order target dynamics and using the concept of attractivity of the manifold based upon the passivity theory results in encompassing various design paradigms under a single roof of the P&I approach. The benchmark examples are solved to show the effectiveness of the proposed P&I approach in confirming how the various design paradigms can be unified.

Golden Riemannian submersions

In this paper, we study a Golden Riemannian submersion between Golden Riemannian manifolds. Here, we investigate the geometric properties of such a submersion and obtain some results. Also, we study the relations between the Ricci curvatures of any fibre, base and target manifolds of Golden Riemannian submersion and using these relations obtain two sharp inequalities. Moreover, we give some characterizations of Golden Riemannian submersion whose total space admits a Ricci soliton with the horizontal or vertical potential field. Finally, we construct some examples of Golden Riemannian submersions.