Mathematical Physics, Analysis and Geometry最新文献

As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur system, two discrete nonlinear Schrödinger type equations are considered. Through the bilinearization reduction method, we construct double Casoratian solutions of the reduced discrete nonlinear Schrödinger type equations, including soliton solutions and Jordan-block solutions. Dynamics of the obtained one-, two-soliton and the simplest Jordan-block solutions are analyzed and illustrated. Moreover, both semi-continuous limit and full-continuous limit, are applied to recover the local and nonlocal semi-discrete nonlinear Schrödinger type equations, as well as the local and nonlocal continuous nonlinear Schrödinger type equations. One-, two-soliton and the simplest Jordan-block solutions for the local and nonlocal semi-discrete nonlinear Schrödinger type equations are constructed and the corresponding dynamics are analyzed and illustrated.

In this paper, we study geometric structures of compact static vacuum spaces and function theoretic properties for the potential functions satisfying the static vacuum equation. In particular, we investigate geometric conditions under which static vacuum spaces are warped product and Bach-flat. As an application, we prove that if a triple ((M^n, g, f), n ge 4), is a compact static vacuum space satisfying (omega :=df wedge i_{nabla f}{mathring{textrm{Ric}}} = 0), then M is either isometric to a round sphere or a warped product of a circle with a compact Einstein manifold of positive Ricci curvature, up to finite cover. Furthermore, if (M, g) has positive isotropic curvature, then M is either isometric to a round sphere or a product ({mathbb {S}}^1 times {mathbb {S}}^{n-1}.)

In earlier joint work with Ruijsenaars, we constructed and studied symmetric joint eigenfunctions (J_N) for quantum Hamiltonians of the hyperbolic relativistic N-particle Calogero–Moser system. For generic coupling values, they are non-elementary functions that in the (N=2) case essentially amount to a ‘relativistic’ generalisation of the conical function specialisation of the Gauss hypergeometric function ({}_2F_1). In this paper, we consider a discrete set of coupling values for which the solution to the joint eigenvalue problem is known to be given by functions (psi _N) of Baker–Akhiezer type, which are elementary, but highly nontrivial, functions. Specifically, we show that (J_N) essentially amounts to the antisymmetrisation of (psi _N) and, as a byproduct, we obtain a recursive construction of (psi _N) in terms of an iterated residue formula.

We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. On an ({text {SU} }(r))-bundle of charge (kappa ) over the 4-sphere, we show that the space of all connections with Yang-Mills energy less than (4 pi ^2 left( |kappa | + 2 right) ) deformation-retracts under Yang-Mills flow onto the space of instantons, allowing us to simplify the proof of Taubes’s path-connectedness theorem. On a compact quaternion-Kähler manifold with positive scalar curvature, we prove that the space of pseudo-holomorphic connections whose (mathfrak {s}mathfrak {p}(1)) curvature component has small Morrey norm deformation-retracts under Yang-Mills flow onto the space of instantons. On a nontrivial bundle over a compact manifold of general dimension, we prove that the infimum of the scale-invariant Morrey norm of curvature is positive.

We extend the theory of quasi-invariant states for compact group actions on (C^{*})-algebras to the setting of semidirect product groups. Given a compact semidirect product (K=Grtimes _{phi }H), where (phi ) is a continuous homomorphism from H into (operatorname {Aut}(G)), we characterize actions of K on (C^{*})-algebras in terms of compatible actions of the component groups G and H. We establish the fundamental properties of K-quasi-invariant states, including cocycle identities, lifting to von Neumann algebras, averaging properties, and prove the main result that under appropriate modular commutation conditions, the GNS representation of a quasi-invariant state is unitarily equivalent to that of its averaged state. This generalizes the framework established by Griseta [3] for single compact groups.

We construct calibrated submanifolds in Euclidean space invariant under the action of a Lie group G. We first demonstrate the method used in this paper by reproducing the results about special Lagrangians in Harvey-Lawson (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982). We then show explicitly that an associative submanifold in (mathbb {R}^7) invariant under the action of a maximal torus (mathbb {T}^2 subset textrm{G}_2) has to be a special Lagrangian submanifold in (mathbb {C}^3). Similarly, we also show that a Cayley submanifold in (mathbb {R}^8) invariant under the action of a maximal torus (mathbb {T}^3 subset text {Spin}(7)) has to be a special Lagrangian submanifold in (mathbb {C}^4). We construct coassociative submanifolds in (mathbb {R}^7) invariant under the action of (textrm{Sp}(1)subset mathbb {H}) with a more general ansatz than the one in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982) but we recover exactly the (textrm{Sp}(1))-invariant coassociatives in (Harvey, R. and Blaine Lawson, H.: Acta Math. 148, 47–157 1982), giving us a rigidity result. Finally, we construct cohomogeneity two examples of coassociative submanifolds in (mathbb {R}^7) which are invariant under the action of a maximal torus (mathbb {T}^2 subset textrm{G}_2).

The initial value spaces of the Painlevé equations are proposed by Okamoto. They are symplectic manifolds in which the Painlevé equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the four dimensional Painlevé system with affine Weyl group symmetry of type ((A_5+A_1)^{(1)}).

We study the spectral properties of the scalar Laplacian on a n-dimensional warped product manifold (M=Sigma times _f N) with a ((n-1))-dimensional compact manifold N without boundary, a one dimensional manifold (Sigma ) without boundary and a warping function (fin C^infty (Sigma )). We consider two cases: (Sigma =S^1) when the manifold M is compact, and (Sigma =mathbb {R}) when the manifold M is non-compact. In the latter case we assume that the warping function f is such that the manifold M has two cusps with a finite volume. In particular, we study the case of the warping function (f(y)=[cosh (y/b)]^{-2nu /(n-1)}) in detail, where (yin mathbb {R}) and b and (nu ) are some positive parameters. We study the properties of the spectrum of the Laplacian in detail and show that it has both the discrete and the continuous spectrum. We compute the resolvent, the eigenvalues, the scattering matrix, the heat kernel and the regularized heat trace. We compute the asymptotics of the regularized heat trace of the Laplacian on the warped manifold M and show that some of its coefficients are global in nature expressed in terms of the zeta function on the manifold N.

In this paper, we extend one of the main results from our joint work [12] with Hone and Mase, in which we studied a deformed type (D_{4}) map, to the general case of type (D_{2N}) for (Nge 3). This can be achieved through a “local expansion" operation, introduced in our joint work [7] with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type (D_{4}) map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type (D_{6}) map and thus enables systematic generalization to higher ranks (D_{2N}). We also study the degree growth of the deformed type (D_{2N}) map via the tropical method and conjecture that, for each N, the deformed map is integrable, as indicated by the algebraic entropy test, a criterion for detecting integrability in discrete dynamical systems.

In this article, we explore the possibility of inheriting an almost Ricci soliton structure on a compact Riemannian manifold (left( M^{n},gright) ) of dimension n through an isometric embedding of (left( M^{n},gright) ) into the Euclidean space (left( R^{m},{overline{g}}right) ), (m>n). For achieving this goal, we choose a constant unit vector (overrightarrow{a}) on (R^{m}) with its tangential component (zeta ) and normal component ({overline{N}}), and call (zeta ) the KN-vector, ({overline{N}}) the KN-normal. We use a lower bound involving a smooth function f on (M^{n}) on the integral of the Ricci curvature (Ricleft( zeta ,zeta right) ) with respect to the KN-vector (zeta ) to show that (left( M^{n},g,zeta ,fright) ) is almost Ricci soliton, which is called the KN-almost Ricci soliton. The mean curvature vector H, gives a natural function (varphi ={overline{g}}left( H,{overline{N}}right) ) on the KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ) called KN-function. Then, we find a condition involving the KN-function (varphi ) to show that an n-dimensional compact proper KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), is isometric to the sphere (S^{n}(c)). In this article, we also find conditions which make a compact KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ) trivial. In first result in this direction, we show that a compact n-dimensional KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), with KN-function (varphi ) and Ricci curvature in the direction of (zeta ) bounded below by (-(n-1)zeta left( varphi right) ) is either isometric to the sphere (S^{n}(c)) or else it is a trivial Ricci soliton. Finally, we show that a compact n-dimensional KN-almost Ricci soliton (left( M^{n},g,zeta ,fright) ), (n>2), having scalar curvature (tau ) and KN-function (varphi ) satisfying (tau varphi ge 0) is necessarily a trivial Ricci soliton.