Annals of Global Analysis and Geometry最新文献

On an asymptotically conical manifold, we prove time decay estimates for the flow of the Schrödinger wave and Klein–Gordon equations via some differentiability properties of the spectral measure. To keep the paper at a reasonable length, we limit ourselves to the low-energy part of the spectrum, which is the one that dictates the decay rates. With this paper, we extend sharp estimates that are known in the asymptotically flat case (see Bouclet and Burq in Duke Math J 170(11):2575–2629, 2021, https://doi.org/10.1215/00127094-2020-0080) to this more general geometric framework and therefore recover the same decay properties as in the Euclidean case. The first step is to prove some resolvent estimates via a limiting absorption principle. It is at this stage that the proof of the previously mentioned authors fails, in particular when we try to recover a low-frequency positive commutator estimate. Once the resolvent estimates are established, we derive regularity for the spectral measure that in turn is applied to obtain the decay of the flows.

We classify all left-invariant pseudo-Riemannian Einstein metrics on (textrm{SL}(2,mathbb {R})times textrm{SL}(2,mathbb {R})) that are bi-invariant under a one-parameter subgroup. We find that there are precisely two such metrics up to homothety, the Killing form and a nearly pseudo-Kähler metric.

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that (mathbb {C}mathbb {P}^3) is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere (mathbb {S}^3). We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on (mathbb {S}^3), whose integral curves are great circles, is a Hopf vector field.

We consider a smooth, complete and non-compact Riemannian manifold ((mathcal {M},g)) of dimension (d ge 3), and we look for solutions to the semilinear elliptic equation

The potential (V :mathcal {M} rightarrow mathbb {R}) is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of (nabla )-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter (lambda ) is sufficiently close to an eigenvalue of the operator (-varDelta _g).

We show the existence of a natural Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the (L^2)-space of the boundary admits Friedrichs extension. We focus on the spectrum of this operator on covering spaces and total spaces of Riemannian principal bundles over compact manifolds.

The description of the Paley–Wiener space for compactly supported smooth functions (C^infty _c(G)) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for (G=textbf{SL}(2,mathbb {R})^d) ((din mathbb {N})) and (G=textbf{SL}(2,mathbb {C})). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

Let X be a compact Thom–Mather stratified pseudomanifold, and let M be the regular part of X endowed with an iterated metric. In this paper, we prove that if the curvature operator of M is bounded, then the (L^2) harmonic space of M is finite dimensional. Next we consider the absolute eigenvalue problems of the Hodge Laplacian of a sequence of compact domains (Omega _j) converging to M. We prove that when the curvature operator of M is bounded, the eigenvalues of (Omega _j) converge to eigenvalues of M, and the eigenforms of (Omega _j) converge to eigenforms of M in the Sobolev norm. This generalizes Chavel and Feldman’s theorem in Chavel and Feldman (J Funct Anal 30:198-222, 1978) from compact manifolds to compact pseudomanifolds and from functions to differential forms. Then, we apply our results to (L^2)-chomology. We will give a correspondence between boundary cohomology and (L^2)-cohomology.