Dimensional Manifold Research Articles

Characterization of a special type of Ricci–Bourguignon soliton on sequential warped product manifold

In this paper, we aim to characterize the sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton. We derive applications of some vector fields like conformal vector field, torse-forming vector field, torqued vector field on [Formula: see text]-almost conformal Ricci–Bourguignon soliton. The inheritance properties of the Einstein-like sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon solitons of class types [Formula: see text] are investigated in this paper. Later we show that if [Formula: see text] is a sequential warped product of a complete connected [Formula: see text]-dimensional Riemannian manifold [Formula: see text] and one-dimensional Riemannian manifolds [Formula: see text] and [Formula: see text] admitting [Formula: see text]-almost conformal Ricci–Bourguignon soliton, then [Formula: see text] becomes a [Formula: see text]-dimensional sphere of radius [Formula: see text]. We have discovered that for a [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton sequential warped product, the warping functions are constants under certain conditions.

Solution to Hessian type equation with prescribed singularities on compact Kähler manifolds

Let (X,ω) be a compact Kähler manifold of dimension n and fix an integer m such that 1≤m≤n. We reformulate Darvas-Nezza-Lu's latest survey [12] into the Hessian setting. Namely, we characterize the relative full mass class Eϕ(X,ω,m) and prove the integration by parts formula of Hessian type. Given a model potential ϕ, we study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. Under some natural conditions on F, we prove that this equation has a unique solution (up to a constant) which has the same singularity type as ϕ.

Chiral Gauge Theory at the Boundary between Topological Phases.

I demonstrate how chiral fermions with an exact gauge symmetry can appear on the d-dimensional boundary of a finite volume (d+1)-dimensional manifold, without any light mirror partners. The condition for the d-dimensional boundary theory to be local is that gauge anomalies cancel and that the volume be large. This can likely be achieved on a lattice and provides a new paradigm for the lattice regularization of chiral gauge theories.

Weyl Fermions on a Finite Lattice.

The phenomenon of unpaired Weyl fermions appearing on the sole 2n-dimensional boundary of a (2n+1)-dimensional manifold with massive Dirac fermions was recently analyzed in D. B. Kaplan [preceding Letter, Chiral gauge theory at the boundary between topological phases, Phys. Rev. Lett. 132, 141603 (2024).PRLTAO0031-900710.1103/PhysRevLett.132.141603]. In this Letter, we show that similar unpaired Weyl edge states can be seen on a finite lattice. In particular, we consider the discretized Hamiltonian for a Wilson fermion in (2+1) dimensions with a 1+1 dimensional boundary and continuous time. We demonstrate that the low lying boundary spectrum is indeed Weyl-like: it has a linear dispersion relation and definite chirality and circulates in only one direction around the boundary. We comment on how our results are consistent with Nielsen-Ninomiya theorem. This work removes one potential obstacle facing the program outlined in D. B. Kaplan, preceding Letter, for regulating chiral gauge theories.

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.

Twisted Dirac Operators and General Kastler–Kalau–Walze Type Theorems for Manifolds with Boundary

In this paper, we establish some general Kastler–Kalau–Walze type theorems on any dimensional manifolds with boundary for twisted Dirac operators.

Some characterizations of Quasi-Einstein and doubly product manifold

This research paper explores the properties of quasi-Einstein manifolds with a unit concircular generator vector field within doubly warped product structures. The paper begins by investigating the characteristics of quasi-Einstein manifolds that possess a unit concircular generator vector field. Subsequently, it analyzes the behavior of the Hessian, Riemann, and Ricci vector fields in the context of a doubly warped product. Moreover, the impact of the quasi-Einstein condition on both the doubly warped product and its factor manifolds is studied. Finally, conditions are derived to determine whether a doubly warped product satisfying the quasi-Einstein condition can be classified as a warped or a direct product. By presenting these findings, this study contributes to the comprehension of quasi-Einstein manifolds and their interactions within doubly warped product structures.

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

Cherrier–Escobar problem for the elliptic Schrödinger-to-Neumann map

In this paper, we study a Cherrier–Escobar problem for the extended problem corresponding to the elliptic Schrödinger-to-Neumann map on a compact 3-dimensional Riemannian manifold with boundary. Using the algebraic topological argument of Bahri and Coron (1988), we show solvability under the assumption that the extended problem corresponding to the elliptic Schrödinger-to-Neumann map has a positive first eigenvalue and a positive Green’s function.

An Interface-Preserving Moving Mesh in Multiple Space Dimensions

An interface-preserving moving mesh algorithm in two or higher dimensions is presented. It resolves a moving ( d -1)-dimensional manifold directly within the d -dimensional mesh, which means that the interface is represented by a subset of moving mesh cell-surfaces. The underlying mesh is a conforming simplicial partition that fulfills the Delaunay property. The local remeshing algorithms allow for strong interface deformations. We give a proof that the given algorithms preserve the interface after interface deformation and remeshing steps. Originating from various numerical methods, data is attached cell-wise to the mesh. After each remeshing operation, the interface-preserving moving mesh retains valid data by projecting the data to the new mesh cells. An open source implementation of the moving mesh algorithm is available at Reference [ 1 ].

Symmetric monoidal equivalences of topological quantum field theories in dimension two and Frobenius algebras

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.

Novel reduction schemes for a dissipative dynamical system: A study on slow invariant manifolds in chemical kinetics

This research explores the intricate concept of the Slow Invariant Manifold (SIM) and its pivotal role in developing model reduction techniques (MRTs) for challenges within dissipative systems in chemical kinetics, specifically in mechanical engineering. Focusing on the multi-step mechanism with two intermediates, primary approximations of the SIM are constructed and compared using two prominent MRTs: The Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). At the given rate coefficient, a special computational experiment was performed in which the efficiency of chemical species has been compared. Noteworthy innovation involves evaluating SIM separately for reduced species, departing from the conventional approach of considering every species within the mechanism. The study employs local sensitivity analysis with MATLAB's Sim-Biology toolbox, presenting quantitative findings in a tabular format for a comprehensive MRT comparison. Beyond contributing to a deeper understanding of model reduction techniques in complex chemical kinetics, this research marks the first computational exploration of dissipative systems. The novel perspective of evaluating SIM for reduced species offers nuanced insights, emphasizing the critical role of SIM in effectively addressing challenges in mechanical engineering applications. In summary, this study introduces computational advancements and novel approaches, advancing model reduction techniques and highlighting the significance of SIM in addressing challenges within mechanical engineering contexts.

Exploratory Data Analysis and Searching Cliques in Graphs

The principal component analysis is a well-known and widely used technique to determine the essential dimension of a data set. Broadly speaking, it aims to find a low-dimensional linear manifold that retains a large part of the information contained in the original data set. It may be the case that one cannot approximate the entirety of the original data set using a single low-dimensional linear manifold even though large subsets of it are amenable to such approximations. For these cases we raise the related but different challenge (problem) of locating subsets of a high dimensional data set that are approximately 1-dimensional. Naturally, we are interested in the largest of such subsets. We propose a method for finding these 1-dimensional manifolds by finding cliques in a purpose-built auxiliary graph.

Asymptotic symmetries of projectively compact order one Einstein manifolds.

We show that the boundary of a projectively compact Einstein manifold of dimension [Formula: see text] can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be identified with asymptotic symmetries of the compactification. The construction is motivated by the investigation of a new curved orbit decomposition for a [Formula: see text] dimensional manifold which we prove results in a line bundle over projectively compact order one Einstein manifolds. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds

Abstract In this paper, we establish compactness results for some classes of conformally compact Einstein metrics defined on manifolds of dimension d ≥ 4. In the special case when the manifold is the Euclidean ball with the unit sphere as the conformal infinity, the existence of such class of metrics has been established in the earlier work of C. R. Graham and J. Lee (“Einstein metrics with prescribed conformal infinity on the ball,” Adv. Math., vol. 87, no. 2, pp. 186–225, 1991). As an application of our compactness result, we derive the uniqueness of the Graham–Lee metrics. As a second application, we also derive some gap theorem, or equivalently, some results of non-existence CCE fill-ins.

Diameter estimates for surfaces in conformally flat spaces

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau’s problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu–Zheng.

Low dimensional approximation and generalization of multivariate functions on smooth manifolds using deep ReLU neural networks

The expressive power of deep neural networks is manifested by their remarkable ability to approximate multivariate functions in a way that appears to overcome the curse of dimensionality. This ability is exemplified by their success in solving high-dimensional problems where traditional numerical solvers fail due to their limitations in accurately representing high-dimensional structures. To provide a theoretical framework for explaining this phenomenon, we analyze the approximation of Hölder functions defined on a d-dimensional smooth manifold M embedded in RD, with d≪D, using deep neural networks. We prove that the uniform convergence estimates of the approximation and generalization errors by deep neural networks with ReLU activation functions do not depend on the ambient dimension D of the function but only on its lower manifold dimension d, in a precise sense. Our result improves existing results from the literature where approximation and generalization errors were shown to depend weakly on D.

Rectifiability of Flat Singular Points for Area-Minimizing mod 2Q Hypercurrents

Abstract Consider an $ m $-dimensional area minimizing mod$ (2Q) $ current $ T $, with $ Q\in {\mathbb {N}} $, inside a sufficiently regular Riemannian manifold of dimension $ m + 1 $. We show that the set of singular density-$ Q $ points with a flat tangent cone is $ (m-2) $-rectifiable. This complements the thorough structural analysis of the singularities of area-minimizing hypersurfaces modulo $ p $ that has been completed in the series of works of De Lellis–Hirsch–Marchese–Stuvard and De Lellis–Hirsch–Marchese–Stuvard–Spolaor, and the work of Minter–Wickramasekera.