Grauert Tubes Research Articles

Riemannian manifolds with entire Grauert tube are rationally elliptic

Riemannian manifolds with entire Grauert tube are rationally elliptic

Poisson and Szegö kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szegö kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order coefficients, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range Oλδ-1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O\\left( \\lambda ^{\\delta -1/2}\\right) $$\\end{document} in all the variables involved, where λ→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\rightarrow +\\infty $$\\end{document} is the asymptotic parameter, rather than rescale according to Heisenberg type.

Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.

An Effective Estimate on Betti Numbers

We provide an effective estimate on the Betti numbers of the loop space of a compact manifold which admits a finite Grauert tube. It implies the polynomial estimate in Chen (arXiv:2101.04368, 2021) after taking the radius of the tube to infinity.

Log-scale equidistribution of nodal sets in Grauert tubes

Let Mτ0 be the Grauert tube (of some fixed radius τ0) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φλ be a Laplacian eigenfunction on M of eigenvalue −λ2 and let φλC be its holomorphic extension to Mτ0. In this article, we prove that on Mτ0∖M, there exists a dimensional constant α>0 and a full density subsequence {λjk}k=1∞ of the spectrum for which the masses of the complexified eigenfunctions φλjkC are asymptotically equidistributed at length scale (log⁡λjk)−α. Moreover, the complex zeros of φλjkC also become equidistributed on this logarithmic length scale.

Left-invariant grauert tubes on SU(2)

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We call such manifolds Grauert tubes, or simply tubes. We consider here the case of M = G a compact connected Lie group with a left-invariant metric, and try to determine for which such metrics the associated tube is entire. It is well-known that the Grauert tube of a bi-invariant metric on a Lie group is entire. The case of the smallest group SU(2) is treated completely, thanks to the complete integrability of the geodesic flow for such a metric, a standard result in classical mechanics. Along the way we find a new obstruction to tubes being entire which is made visible by the complete integrability. (New reference and exposition shortened, 11/17/2017.)

Local and global analysis of nodal sets

This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are `global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, giving upper and lower bounds for hypersurface measures of nodal sets in the setting of general $C^{\infty}$ metrics. The proofs of both their polynomial upper bound and sharp lower bound are sketched. The survey continues with a global proof of the sharp upper bound for real analytic metrics (originally proved by Donnelly-Fefferman with local arguments), using analytic continuation to Grauert tubes. Then it reviews results of Toth-Zelditch giving sharp upper bounds on Hausdorff measures of intersections of nodal sets with real analytic submanifolds in the real analytic setting. Last, it goes over lower bounds of Jung-Zelditch on numbers of nodal domains in the case of $C^{\infty}$ surfaces of non-positive curvature and concave boundary or on negatively curved `real' Riemann surfaces, which are based on ergodic properties of the geodesic flow and eigenfunctions, and on estimates of restrictions of eigenfunctions to hypersurfaces. The last section details recent results on restrictions.

The Poisson transform on a compact real analytic Riemannian manifold

We study the transform mapping an \(L^{2}\) function f on a compact, real analytic Riemannian manifold X to the analytic continuation of \(\exp (-t \sqrt{\Delta })f\) to the interior of a Grauert tube tube \(M_{t}\) about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from \(L^{2}(X)\) onto the holomorphic \(L^{2}\) functions on \(M_{t}\). If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restrict ion principle.

HIGHER ORDER SCHWARZIANS FOR GEODESIC FLOWS, MOMENT SEQUENCES AND THE RADIUS OF ADAPTED COMPLEXIFICATIONS

In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical theorem of Loewner on analytic continuation we are able to characterize the existence of the adapted complex structure induced by g on the set T^RM of vectors in TM of length up to R, equivalently for M compact, to the existence of a Grauert tube of radius R in terms of infinite Hankel matrices involving these Schwarzian functions. The basic characterization so obtained can be expressed as a sequence of differential inequalities of increasing order polynomial in the covariant derivatives of the Gauss curvature on M and in {\pi}/R that should be regarded as the higher order versions of a curvature inequality by L. Lempert and R. Sz\oke. The second part of the paper, sections 7 through 11, includes a discussion of the rank of the infinite Hankel matrix of the Schwarzians from part 1 and of new Schwarzians defined now for purely imaginary radius, as well as some computations and examples. A characterization of the existence of the adapted structure on T^RM in terms of moment sequences with parameters R and v in TM is also noted.

The Tian–Yau–Zelditch Theorem and Toeplitz Operators

AbstractZelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

THE ADAPTED COMPLEXIFICATION OF AN ELLIPSOID

We show that an n-dimensional real ellipsoid in ℝn+1 with the induced Riemannian metric does not admit an unbounded adapted complexification in the sense of Lempert/Szőke and Guillemin/Stenzel, unless it is a round sphere. In other words, an ellipsoid whose (maximal) Grauert tube has infinite radius must be a round sphere. For the proof we take advantage of the integrability of the geodesic flow and use a classical theorem on umbilic geodesics. We carry out an extension of this result to Liouville metrics elsewhere.

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φλ ℂ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If $\{\phi_{j_{k}}\}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g$ , where $[Z_{\phi_{j_k}^{\mathbb{C}}}]$ is the current of integration over the complex zeros and where $\overline{\partial}$ is with respect to the adapted complex structure of Lempert-Szöke and Guillemin-Stenzel.

The logarithmic term of the Szegö kernel for two-dimensional Grauert tubes

Every compact real-analytic Riemannian manifold has a complexification called the Grauert tube. We give an asymptotic expansion of the leading coefficient of the logarithmic term of the Szego kernel for two-dimensional Grauert tubes. where and are functions on which are smooth up to , and ( ) is a defin- ing function for with ( ) 0 in . We note that the leading coeffici ent of the log- arithmic term is independent of the choice of ( ) and gives a pseudo-hermitian invariant of . More precisely speaking, it can be written as a linear combination of complete contractions, with respect to the Levi-form, of the tensor products of the Tanaka-Webster curvature, torsion and their covariant derivatives. In case = 2, an explicit form of was given in (5).

On rigidity of Grauert tubes over homogeneous Riemannian manifolds

Given a real-analytic Riemannian manifold $X$ there is a canonical complex structure, which is compatible with the canonical complex structure on $T^*X$ and makes the leaves of the Riemannian foliation on $TX$ into holomorphic curves, on its tangent bundle. A {\it Grauert tube} over $X$ of radius $r$, denoted as $T^rX$, is the collection of tangent vectors of $X$ of length less than $r$ equipped with this canonical complex structure. In this article, we prove the following two rigidity property of Grauert tubes. First, for any real-analytic Riemannian manifold such that $r_{max}>0$, we show that the identity component of the automorphism group of $T^rX$ is isomorphic to the identity component of the isometry group of $X$ provided that $r<r_{max}$. Secondly, let $X$ be a homogeneous Riemannian manifold and let the radius $r<r_{max}$, then the automorphism group of $T^rX$ is isomorphic to the isometry group of $X$ and there is a unique Grauert tube representation for such a complex manifold $T^rX$.

The geometry of Grauert tubes and complexification of symmetric spaces

We consider complexifications of Riemannian symmetric spaces $X$ of nonpositive curvature. We show that the maximal Grauert domain of $X$ is biholomorphic to a maximal connected extension $\Omega\sb {{\rm AG}}$ of $X=G/K\subset G\sb {\mathbb {C}}/K\sb {\mathbb {C}}$ on which $G$ acts properly, a domain first studied by D. Akhiezer and S. Gindikin [1]. We determine when such domains are rigid, that is, when ${\rm Aut}\sb {\mathbb {C}}(\Omega\sb {{\rm AG}}=G$ and when it is not (when \Omega\sb {{\rm AG}}$ has hidden symmetries). We further compute the $G$-invariant plurisubharmonic functions on $\Omega\sb {{\rm AG}}$ and related domains in terms of Weyl group invariant strictly convex functions on a $W$-invariant convex neighborhood of $0\in \mathfrak {a}$. This generalizes previous results of M. Lassalle [25] and others. Similar results have also been proven recently by Gindikin and B. Krotz [8] and by Krotz and R. Stanton [24].

Complexifications of nonnegatively curved manifolds

Define a good complexification of a closed smooth manifold M to be a smooth affine algebraic variety U over the real numbers such that M is diffeomorphic to U(R) and the inclusion U(R) → U(C) is a homotopy equivalence. Kulkarni showed that every manifold which has a good complexification has nonnegative Euler characteristic [16]. We strengthen his theorem to say that if the Euler characteristic is positive, then all the odd Betti numbers are zero. Also, if the Euler characteristic is zero, then all the Pontrjagin numbers are zero (see Theorem 1.1 and, for a stronger statement, Theorem 2.1). We also construct a new class of manifolds with good complexifications. As a result, all known closed manifolds which have Riemannian metrics of nonnegative sectional curvature, including those found by Cheeger [5] and Grove and Ziller [11], have good complexifications. We can in fact ask whether a closed manifold has a good complexification if and only if it has a Riemannian metric of nonnegative sectional curvature. The question is suggested by the work of Lempert and Szőke [17]. Lempert and Szőke, and independently Guillemin and Stenzel [12], constructed a canonical complex analytic structure on an open subset of the tangent bundle of M, given a real analytic Riemannian metric on M. We say that a real analytic Riemannian manifold has entire Grauert tube if this complex structure is defined on the whole tangent bundle TM. Lempert and Szőke found that every Riemannian manifold with entire Grauert tube has nonnegative sectional curvature. Also, a conjecture by Burns [4] predicts that for every closed Riemannian manifold M with entire Grauert tube, the complex manifold TM is an affine algebraic variety in a natural way; if this is correct, the complex manifold TM would be a good complexification of M in the above sense. The problem remains open at this writing despite the paper of Aguilar and Burns [2], because the proof of Proposition 2.1 there is not yet generally accepted. We can also ask how many Riemannian manifolds have entire Grauert tube. It is known to be a strong restriction: Szőke showed that of all surfaces of revolution diffeomorphic to the 2-sphere, only a one-parameter family of metrics of given volume have entire Grauert tube [22]. Aguilar showed that the quotient of a Riemannian manifold with entire Grauert tube by a group of isometries acting freely also has entire Grauert tube [1]. All known manifolds with entire Grauert

Orbits of the geodesic flow and chains on the boundary of a Grauert tube

A \(C^{\omega}\) Riemannian manifold (X,g) determines an integrable complex structure on a tubular neighborhood, \(T^{*\epsilon}X\), of the zero section in \(T^{*}X\) and a CR-structure on the boundary, \(M^{\epsilon}\). There are two natural families of curves on \(M^{\epsilon}\): the orbits of the geodesic flow and a CR-invariant family called chains. It is natural to ask whether they are related. We show that if orbits of the geodesic flow are chains on \(M^{\epsilon}\) for all \(\epsilon\) sufficiently small, then (X,g) is Einstein. As a partial converse we show that if (X,g) is harmonic, then orbits of the geodesic flow are chains. To prove this we study the Fefferman metric associated with \(M^{\epsilon}\).

A family of Kähler-Einstein manifolds and metric rigidity of Grauert tubes

In this paper we explain how the so-called adapted complex structures can be used to associate to each compact real-analytic Riemannian manifold a family of complete Kahler-Einstein metrics and show that already one element of this family uniquely determines the original manifold. The underlying manifolds of these metrics are open disc bundles in the tangent bundle of the original Riemannian manifold. Recall first the notion of adapted complex structures (cf. [7], [12], [16]). Let (M, g) be a complete real-analytic Riemannian manifold. If γ is a geodesic in M , we can define a map ψγ : C→ TM by ψγ : σ + iτ 7→ τ γ(σ). If r ∈ (0,∞], we let T M denote the open disc bundle in TM consisting of tangent vectors of norm less than r (note that we allow r to be infinite). A complex structure on T M is said to be adapted with respect to g if ψγ is holomorphic on ψ−1 γ (T M) for each geodesic γ. We shall usually omit the phrase “with respect to” if it is obvious which metric is being discussed. The manifolds T M are also called Grauert tubes since Grauert used such manifolds in his famous result to show that each real-analytic manifold admits a real-analytic embedding to a euclidean space. Adapted complex structures were discovered by studying certain global solutions of the complex homogeneous Monge-Ampere equation on Stein manifolds (cf. [2], [7], [12], [14]). Their basic properties were treated in [7], [12], [16]. Among others, one has an existence result: if (M, g) is a compact real-analytic Riemannian manifold, then there exists an r ∈ (0,∞] such that T M carries an adapted complex structure [7], [16]. Also the adapted complex structure is uniquely determined by (M, g) ([7], [12]). From now on we shall take (M, g) to be a compact Riemannian manifold and use R to denote the largest element of (0,∞] such that TM supports an adapted complex structure. It was shown in [12] that the energy function E on TM which assigns to each tangent vector the half of its norm-square with respect to the metric g is strictly plurisubharmonic on TM . Thus another theorem of Grauert implies that TM is a Stein manifold. Hence T M is relatively compact and strictly pseudoconvex whenever 0 < r < R. Received by the editors February 2, 2000. 2000 Mathematics Subject Classification. Primary 32Q15, 53C35.

Symplectic geometry and the uniqueness of Grauert tubes

A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can arise in more than one way from this construction. We show that given a compact M and a finite exhaustion, the underlying Riemannian structure is unique. The proof uses the technique of holomorphic disks spanning two exact Lagrangian submanifolds of the cotangent bundle of M, and Schwarz reflection.

Distribution of Resonances for Asymptotically Euclidean Manifolds

In this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds, a class of manifolds generalizing Euclidian n-space. Subject to the basic assumption of analyticity near infinity, we show that resolvent of the Laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum. This involves a geometric interpretation of the complex scaling method in terms of deformations in the Grauert tube of the manifold. We then show that the number of resonances (poles of the meromorphic continuation of the resolvent) in a conic neighborhood of $\mathbb{R}_+$of absolute value less than $r^2$ is $\mathcal O(r^n)$. Under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow, we prove a finer, Weyl-type upper bound for the counting function for resonances in small neighborhoods of the real axis. This estimate has an exponent which involves the dimension of the trapped set of the geodesic flow.