Annals of Global Analysis and Geometry最新文献

In this paper we establish a comparison formula of the absolute and relative real analytic torsion forms over fibrations with boundaries. The key tool is a gluing formula of analytic torsion forms proved by Puchol and Zhang and the author. As a consequence of the comparison formula, we prove another version of the gluing formula of the analytic torsion forms conjectured by the author.

As an application of the general theory on extrinsic geometry (Doubrov et al. in SIGMA Symmetry Integr Geom Methods Appl 17:061, 2021), we investigate extrinsic geometry in flag varieties and systems of linear PDE’s for a class of special interest associated with the adjoint representation of (mathfrak {sl}(3)). We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE’s of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE’s called contact Cayley’s surface and one which has (varvec{mathfrak {sl}}(2)) symmetry.

For a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space ({{mathcal {S}}}(Kbackslash G/K)) isomorphically onto the space ({{mathcal {S}}}(Sigma _{{mathcal {D}}})), where (Sigma _{{mathcal {D}}}) is an embedded copy of the Gelfand spectrum in ({{mathbb {R}}}^ell ), canonically associated to a generating system ({{mathcal {D}}}) of G-invariant differential operators on G/K, and ({{mathcal {S}}}(Sigma _{{mathcal {D}}})) consists of restrictions to (Sigma _{{mathcal {D}}}) of Schwartz functions on ({{mathbb {R}}}^ell ). Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair ((M_n,SO_n)) with (n=3,4). The rather trivial case (n=2) is included in previous work by the same authors.

We construct the hyper-Kähler moduli space of framed monopoles over (mathbb {R}^3) for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence arbitrary symmetry breaking. In order to do so, we define a configuration space of pairs with appropriate asymptotic conditions and perform an infinite-dimensional quotient construction. We make use of the b and scattering calculuses to study the relevant differential operators.

We show that if the universal cover of a closed smooth manifold admitting a metric with non-negative Ricci curvature is formal, then the manifold itself is formal. We reprove a result of Fiorenza–Kawai–Lê–Schwachhöfer, that closed orientable manifolds with a non-negative Ricci curvature metric and sufficiently large first Betti number are formal. Our method allows us to remove the orientability hypothesis; we further address some cases of non-closed manifolds.

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds.

In this paper, we prove a stability result for the non-Kähler geometry of locally conformally Kähler (lcK) spaces with singularities. Specifically, we find sufficient conditions under which the image of an lcK space by a holomorphic mapping also admits lcK metrics, thus extending a result by Varouchas about Kähler spaces.

Let (G=left( V,Eright) ) be a connected finite graph. We are concerned about the Kazdan–Warner equation in the negative case on G, say

where (Delta ) is the graph Laplacian, (c<0) is a real constant, (h_lambda =h+lambda ), (h:Vrightarrow mathbb {R}) is a function satisfying (hle max _{V}h=0) and (hnot equiv 0), (lambda in mathbb {R}). In this paper, using the method of topological degree, we prove that there exists a critical value (Lambda ^*in (0,-min _{V}h)) such that if (lambda in (-infty ,Lambda ^*]), then the above equation has solutions; and that if (lambda in (Lambda ^*,+infty )), then it has no solution. Specifically, if (lambda in (-infty ,0]), then it has a unique solution; if (lambda in (0,Lambda ^*)), then it has at least two distinct solutions, of which one is a local minimum solution; while if (lambda =Lambda ^*), it has at least one solution. For the proof of these results, we first calculate the topological degree of a map related to the above equation, and then we utilize the relationship between the topological degree and the critical group of the relevant functional. Our method is essentially different from that of Liu and Yang (Calc. Var. 59 (2020), 164), who obtained similar results by using a method of variation.

We classify compact self-dual almost-Kähler four-manifolds of positive type and zero type. In particular, using LeBrun’s result, we show that any self-dual almost-Kähler metric on a manifold which is diffeomorphic to ({{mathbb {C}}}{{mathbb {P}}}_{2}) is the Fubini-Study metric on ({{mathbb {C}}}{{mathbb {P}}}_{2}) up to rescaling. In case of negative type, we classify compact self-dual almost-Kähler four-manifolds with J-invariant ricci tensor.