Annals of Global Analysis and Geometry最新文献

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.

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list several properties of such foliations and propose two conjectures about the topological types of their leaves, both of which coincide with the classical Weinstein conjecture in the case of contact flows. We give positive partial results for our conjectures in particular cases—when the holonomy of the contact foliation preserves a Riemannian metric, for instance—extending already established results in the field of Contact Dynamics.

Given a constant C and a smooth closed ((n-1))-dimensional Riemannian manifold ((Sigma , g)) equipped with a positive function H, a natural question to ask is whether this manifold can be realised as the boundary of a smooth n-dimensional Riemannian manifold with scalar curvature bounded below by C and boundary mean curvature H. That is, does there exist a fill-in of ((Sigma ,g,H)) with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author (Int Math Res Not 7:2019, 2019) to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.

We present straightforward conditions which ensure that a strongly elliptic linear operator L generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that L is ‘sectorial,’ a condition that specifies the decay of the resolvent ((lambda I - L)^{-1}) as (lambda ) diverges from the Hölder spectrum of L. A key step is that we prove existence of this resolvent if (lambda ) is sufficiently large using a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and (e^{-tL}) we obtain can then be used to prove well-posedness of a wide class of nonlinear flows. We illustrate this by proving well-posedness on Hölder spaces of the flow associated with the ambient obstruction tensor on complete manifolds of bounded geometry. This new result for a higher-order flow on a noncompact manifold exhibits the broader applicability of our technique.