Compact Homogeneous Manifolds Research Articles

On anti-quasi-Sasakian manifolds of maximal rank

We discuss the existence of invariant anti-quasi-Sasakian (aqS) structures of maximal rank on compact homogeneous Riemannian manifolds and on nilpotent Lie groups. In the former case we obtain a non-existence result, while in the latter case we provide a complete classification. We also show that every compact aqS manifold has nonvanishing second Betti number.

Vertex operator for generalized Kac–Moody algebras associated to the two-sphere and the two-torus

We pursue our study of generalized Kac–Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realizations of the semi-direct product of the extension of Kac–Moody and Virasoro algebras on [Formula: see text] and [Formula: see text], respectively. As for the fermionic realization previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularize infinite sums by means of the Riemann [Formula: see text]-function.

Polynomial mixing for time-changes of unipotent flows

Let $G$ be a connected semisimple Lie group with finite centre, and let $M= \Gamma \backslash G$ be a compact homogeneous manifold. Under a spectral gap assumption, we show that smooth time-changes of any unipotent flow on $M$ have polynomial decay of correlations. Our result applies also in the case where $M$ is a finite volume, non-compact quotient under some additional assumptions on the generator of the time-change. This generalizes a result by Forni and Ulcigrai (JMD, 2012) for smooth time-changes of horocycle flows on compact surfaces.

Determinants and Plemelj-Smithies formulas

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincaré type determinant for operators on the torus {{mathbb T}^n} and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

Widths and entropy of sets of smooth functions on compact homogeneous manifolds

Widths and entropy of sets of smooth functions on compact homogeneous manifolds

Complex Geometry of Iwasawa Manifolds

Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

On the nonexistence of S6 type complex threefolds in any compact homogeneous complex manifolds with the compact lie group G2 as the base manifold

On the nonexistence of S6 type complex threefolds in any compact homogeneous complex manifolds with the compact lie group G2 as the base manifold

Hausdorff–Young inequality for Orlicz spaces on compact homogeneous manifolds

We prove the classical Hausdorff–Young inequality for Orlicz spaces on compact homogeneous manifolds.

On the nuclear trace of Fourier integral operators

In this paper we characterise ther-nuclearity of Fourier integraloperators on Lebesgue spaces. Fourier integral operators will be consideredinRn,the discrete groupZn,then-dimensional torus and symmetric spaces(compact homogeneous manifolds). We also give formulae forthe nucleartrace of these operators. Explicit examples will be given onZn,the torusTn,the special unitary group SU(2),and the projective complex planeCP2.Ourmain theorems will be applied to the characterization ofr-nuclear pseudo-differential operators defined by the Weyl quantization procedure.

Hermitian Curvature Flow on Compact Homogeneous Spaces

We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite extinction time $$T>0$$ and we analyze its behavior when $$t\rightarrow T$$ . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [17] on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in Lp(SU(2)) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the Lp-Lq boundedness of Fourier multipliers for 1<p≤2≤q<∞ on compact homogeneous manifolds as well as the Lp-Lq boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby-Wang fibration over complex flag manifolds as well as their induced homogeneous Sasaki-Einstein structures. As an application of our results we use the Cartan-Remmert reduction and the Calabi ansatz technique to provide many explicit examples of crepant resolutions of Calabi-Yau cones with certain homogeneous Sasaki-Einstein manifolds realized as links of isolated hypersurface singularities.

Titchmarsh theorems for Fourier transforms of H\xf6lder\u2013Lipschitz functions on compact homogeneous manifolds

In this paper we extend classical Titchmarsh theorems on the Fourier transform of Hölder–Lipschitz functions to the setting of compact homogeneous manifolds. As an application, we derive a Fourier multiplier theorem for L^2-Hölder–Lipschitz spaces on compact Lie groups. We also derive conditions and a characterisation for Dini–Lipschitz classes on compact homogeneous manifolds in terms of the behaviour of their Fourier coefficients.

Mostow’s fibration for canonical embeddings of compact homogeneous CR manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Mostow’s fibration for canonical embeddings of compact homogeneous $CR$ manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

SPIN STRUCTURES ON COMPACT HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

On the automorphism group of a closed G2-structure

We study the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel G$_2$-structure, showing that its identity component is abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G$_2$-structure. We also discuss some relevant examples.

Quasi-periodic solutions to nonlinear beam equations on compact Lie groups with a multiplicative potential

The goal of this work is to study the existence of quasi-periodic solutions to nonlinear beam equations with a multiplicative potential. The nonlinearity is required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogeneous manifold with respect to a compact Lie group, which includes standard torus Td, special orthogonal group SO(d), special unitary group SU(d), spheres Sd and the real and complex Grassmannians. The proof is based on a differentiable Nash–Moser iteration scheme.