Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension ${2n-1, n\,\geqslant\,2}$ , and let L ^k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L ^k , for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.     

Szegő kernel expansion and equivariant embedding of CR manifolds with circle action

Let X be a compact strongly pseudoconvex CR manifold with a transversal CR \(S^1\)-action. In this paper, we establish the asymptotic expansion of Szeg? kernels of positive Fourier components, and by using the asymptotics, we show that X can be equivariant CR embedded into some \(\mathbb {C}^N\) equipped with a simple \(S^1\)-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.     

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds

In this paper,we discuss a Kazdan-Warner typed equation on certain non-compact Rie- mannian manifolds.As an application,we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundles over certain non-compact K(?)hler manifolds.     

The Szegö function on a non-rectifiable arc

Let Γ be a simple Jordan arc in the complex plane. The Szegö function, by definition, is a holomorphic in ? Γ function with a prescribed product of its boundary values on Γ. The problem of finding the Segö function in the case of piecewise smooth Γ was solved earlier. In this paper we study this problem for non-rectifiable arcs. The solution relies on properties of the Cauchy transform of certain distributions with the support on Γ.     

Explicit construction of a Chern–Moser connection for CR manifolds of codimension two

In the present paper we suggest an explicit construction of a Cartan connection for an elliptic or hyperbolic CR manifold M of dimension six and codimension two, i.e. a pair , consisting of a principal bundle over M and of a Cartan connection form over P, satisfying the following property: the (local) CR transformations are in one to one correspondence with the (local) automorphisms for which . For any , this construction determines an explicit monomorphism of the stability subalgebra Lie (Aut(M)_x) into the Lie algebra of the structure group H of P. Mathematics Subject Classification (2000) Primary 32V05, Secondary 53C15, 53A55     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

The energy of a Kähler class on admissible manifolds

On a compact complex manifold of Kähler type, the energy E(Ω) of a Kähler class Ω is given by the squared L ²-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kähler metric representing the said class, and any one such metric whose scalar curvature has squared L ²-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kähler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kähler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kähler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.     

On the asymptotics of the on-diagonal Szegö kernel of certain Reinhardt domains

We compute the leading and subleading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.     

The $$L^p$$ L p -Calderón–Zygmund inequality on non-compact manifolds of positive curvature

Annals of Global Analysis and Geometry - We construct, for $$p>n$$ , a concrete example of a complete non-compact n-dimensional Riemannian manifold of positive sectional curvature which does...     

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

Semi-classical resolvent estimates for the Schrödinger operator on non-compact complete Riemannian manifolds

We prove uniform semi-classical estimates for the resolvent of the Schrödinger operator h ² _g + V (x), 0 < h 1, at a nontrapping energy level E > 0, where V is a real-valued non-negative potential and _g denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold which may have a nonempty compact smooth boundary.*Partially supported by CNPq (Brazil)     

The L^2-Atiyah–Bott–Lefschetz theorem on manifolds with conical singularities: a heat kernel approach

Using an approach based on the heat kernel, we prove an Atiyah–Bott–Lefschetz theorem for the $L^2$ -Lefschetz numbers associated with an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.     

The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

Consequences of the existence of a non-compact operator between nuclear Köthe spaces

It is proved that if a continuous, linear, non-compact operator TnK(A)K(B) exists between two nuclear Köthe spaces then there exists a common (up to isomorphism) step space of the two spaces provided K(A) is regular and K(B) is isomorphic to a subspace of K(A) or K(B) is regular and K(A) is isomorphic to a quotient space of K(B) or K(A) and K(B) satisfy a certain splitting condition. Consequences in some particular cases are also obtained.