Annals of Global Analysis and Geometry最新文献

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.

We prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of (G_2) data. These formulae highlight the role of the ambient torsion and Ricci curvature. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds. These results apply, for example, to coassociative fibrations. We illustrate our formulae with several examples, both homogeneous and non.

In this work, we will study the boundary behaviors of a spacelike positive constant mean curvature surface (Sigma ) in the Schwarzschild spacetime exterior to the black hole. We consider two boundaries: the future null infinity (mathcal {I}^+) and the horizon. Suppose near (mathcal {I}^+), (Sigma ) is the graph of a function (-P(textbf{y},s)) in the form (overline{v}=-P), where (overline{v}) is the retarded null coordinate with (s=r^{-1}) and (textbf{y}in mathbb {S}^2). Suppose the boundary value of (P(textbf{y},s)) at (s=0) is a smooth function f on the unit sphere (mathbb {S}^2). If P is (C^4) at (mathcal {I}^+), then f must satisfy a fourth order PDE on (mathbb {S}^2). If P is (C^3), then all the derivatives of P up to order three can be expressed in terms of f and its derivatives on (mathbb {S}^2). For the extrinsic geometry of (Sigma ), under certain conditions we obtain decay rate of the trace-free part of the second fundamental forms (mathring{A}). In case (mathring{A}) decays fast enough, some further restrictions on f are given. For the intrinsic geometry, we show that under certain conditions, (Sigma ) is asymptotically hyperbolic in the sense of Chruściel–Herzlich (Pac J Math 212(2):231–264, 2003). Near the horizon, we prove that under certain conditions, (Sigma ) can be expressed as the graph of a function u which is smooth in (eta =left( 1-frac{2m}{r}right) ^{frac{1}{2}}) and (textbf{y}in mathbb {S}^2), and all its derivatives are determined by the boundary value u at (eta =0). In particular, a Neumann-type condition is obtained. This may be related to a remark of Bartnik (in: Proc Centre Math Anal Austral Nat Univ, 1987). As for intrinsic geometry, we show that under certain conditions the inner boundary of (Sigma ) given by (eta =0) is totally geodesic.

We prove some vanishing theorems for the Kohn–Rossi cohomology of some spherical CR manifolds. To this end, we use a canonical contact form defined via the Patterson–Sullivan measure and Weitzenböck-type formulae for the Kohn Laplacian. We also see that our results are optimal in some cases.

We discuss in detail Alan Schoen’s I-WP surface, an embedded triply periodic minimal surface of genus 4 with cubical symmetries. We exhibit various geometric realizations of this surface with the same conformal structure and use them to prove that the associate family of the I-WP surface contains six surfaces congruent to I-WP at Bonnet angles that are multiples of (60^circ ).

We show that under some natural geometric assumption, Einstein metrics on conformal products of two compact conformal manifolds are warped product metrics.