Canonical Metric Research Articles

Invariant differential operators associated with a conformal metric

Peschl defined invariant higher-order derivatives of a holomorphic or meromorphic function on the unit disk. Here, the invariance is concerned with the hyperbolic metric of the source domain and the canonical metric of the target domain. Minda and Schippers extended Peschl’s invariant derivatives to the case of general conformal metrics. We introduce similar invariant derivatives for smooth functions on a Riemann surface and show a complete analogue of Faa di Bruno’s formula for the composition of a smooth function with a holomorphic map with respect to the derivatives. An interpretation of these derivatives in terms of intrinsic geometry and some applications will be also given.

Energy functionals and canonical Kähler metrics

Yau [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the Ek functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional E1 is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that E1 is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals Ek are bounded below on the space of metrics with nonnegative Ricci curvature

Affine differential geometry of the unit normal vector fields of hypersurfaces in the real space forms

In this paper, for a hypersurface in the real space form of constant curvature, we prove that the unit normal vector field is an affine imbedding into a certain sphere bundle with canonical metric. Moreover, we study the relations between a hypersurface and its unit normal vector field as an affine imbedding. In particular, several hypersurfaces are characterized by affine geometric conditions which are independent of the choice of the transversal bundle.

Bounds on canonical Green's functions

A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

Space-time matter inflation

We study a model of power-law inflationary inflation using the space-time-matter theory of gravity for a five-dimensional canonical metric that describes an apparent vacuum. In this approach the expansion is governed by a single scalar (neutral) quantum field. In particular, we study the case where the power of expansion of the universe is p⪢1. This kind of model is more successful than others in accounting for galaxy formation.

Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

The Frechet manifold of all embeddings (up to orientation preserving reparametrizations) of the circle in S 3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S 3 to prove that among all such fibrations p:S 3?B, the manifold B consisting of the oriented fibers is totally geodesic in , or has minimum volume or diameter with the induced metric, exactly when p is a Hopf fibration

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μt be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between μt and μ is not small for t ≪λ−1 logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

A.A. Kirillov has given a parametrization of the space U ∞ of univalent functions on the closed unit disk, which are C ∞ up to the boundary, by Diff ( S 1 ) / S 1 where Diff ( S 1 ) denotes the group of orientation preserving diffeomorphisms of the circle S 1 . In the same spirit, the space J ∞ of C ∞ Jordan curves in the complex plane can be parametrized by the double quotient SU ( 1 , 1 ) \\ Diff ( S 1 ) / SU ( 1 , 1 ) . As a consequence, J ∞ carries a canonical Riemannian metric. We construct a canonical Brownian motion on U ∞ . Classical technologies of the theory of univalent functions, like Beurling–Ahlfors extension, Loewner equation, Beltrami equation, developed in the context of Kunita's stochastic flows, are the tools for obtaining this result which should be seen as a first step to the construction of a canonical Brownian motion on J ∞ .

Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ? 2

Let Ω be a bounded symmetric domain of rank ≥ 2, and Γ ⊂ Aut(Ω) be a torsion-free irreducible cocompact lattice, X := Ω/Γ. On the projective manifold X there is the canonical Kahler-Einstein metric, which is of nonpositive holomorphic bisectional curvature. In Mok [M1,2] we established a Hermitian metric rigidity theorem for such projective manifolds X, which in the case when Ω is irreducible says that any Hermitian metric of nonpositive curvature in the sense of Griffiths is necessarily a constant multiple of the Kahler-Einstein metric. As a consequence, we proved in the latter case that any nontrivial holomorphic mapping f : X → Z into a Hermitian manifold Z of nonpositive curvature in the sense of Griffiths is necessarily an isometric immersion totally geodesic with respect to the Hermitian connection on Z. The Hermitian metric rigidity theorem can be taken as a tool in establishing the statement that any f : X → Z is necessarily a holomorphic immersion, a statement which only concerns the complex structure of X and Z. The Hermitian metric rigidity theorem and its consequences were generalized by To [To] to be applicable to any torsion-free lattice Γ ⊂ Aut(Ω), where X := Ω/Γ may be noncompact.

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.

On a Multi-particle Moser-Trudinger Inequality

We verify a conjecture of Gillet-Soule. We prove that the determinant of the Laplacian on a line bundle over CP is always bounded from above. This can also be viewed as a multiparticle generalization of the Moser-Trudinger Inequality. Furthermore, we conjecture that this functional achieves its maximum at the canonical metric. We give some evidence for this conjecture, as well as links to other fields of analysis.

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out.

On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains

Let Ω be a bounded symmetric domain equipped with a canonical Kahler metric. We are interested to characterize holomorphic geodesic cycles (i.e., compact complex geodesic submanifolds) S ⊂ X on Hermitian locally symmetric manifolds X uniformized by Ω, i.e., X = Ω/Γ, where Γ ⊂ Aut(Ω) is any torsion-free discrete group of automorphisms, in terms of differentialgeometric or algebro-geometric conditions. In Eyssidieux-Mok [EysMok1995] we studied almost geodesic complex submanifolds and formulated the gap phenomenon. Up to equivalence under Aut(Ω) there are only a finite number of totally geodesic complex submanifolds D ⊂ Ω which are themselves biholomorpic to bounded symmetric domains. Let 2 > 0. We say that S ⊂ X is 2-geodesic if and only if the norm of the second fundamental form of S ⊂ X is uniformly bounded by 2. Fixing Ω but letting Γ ⊂ Ω be arbitrary we showed that when 2 is sufficiently small, an 2-geodesic compact complex

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold OHgr(Gamma)/Gamma, where Gamma is a convex cocompact subgroup of AutCRS2n+1=PU(n+1,1) and OHgr(Gamma) is the discontinuity domain of Gamma. This contact form can be used to prove that OHgr(Gamma)/Gamma is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent delta(Gamma)n, or delta(Gamma)=n). This generalizes Nayatanirsquos result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let {\rm M} = \Gamma \backslash X be a quotient of X by a torsion-free discrete subgroup \Gamma of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a (G_{\Bbb C} , X_c) -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group {\rm G}_{\Bbb C} . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ({\rm G}_{\Bbb C} , X_c) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.

Induced Matter and Particle Motion in Non-Compact Kaluza-Klein Gravity

We examine generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate in the four-dimensional metric. We discuss a more appropriate way to interpret the four-dimensional energy-momentum tensor induced from the five-dimensional space-time and show it can lead to quite different physical situations depending on the interpretation chosen. Furthermore, we show that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, we show the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force. In the examples given, we show that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario.

Affine spheres and convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-manifolds

We outline consequences of a theorem of Cheng-Yau in affine differential geometry for manifolds locally modeled on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. In particular, for properly convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-manifolds, we observe that there are two canonical projectively flat connections and a canonical Riemannian metric. One connection represents the given [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-structure and the other the [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]-structure of the projective dual manifold. When n is 2, we use this approach together with a result of C.-P. Wang to show that a compact oriented convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /]-surface of genus at least two is equivalent to a conformal structure on the surface together with a holomorphic tricanonical form. This recovers a theorem of Goldman on the deformation space of such surfaces, and yields a description of the moduli space.

The canonical connection of a bi-Lagrangian manifold

We prove that the canonical connection of a bi-Lagrangian manifold introduced by Hess is a Levi-Civita connection, showing that a bi-Lagrangian manifold (i.e. a symplectic manifold endowed with two transversal Lagrangian foliations) is endowed with a canonical semi-Riemannian metric.

Analogs of Wiener's ergodic theorems for semisimple Lie groups. II

Let G be a connected finite-center simple Lie group of real rank one, and let K be a maximal compact subgroup of G. Let A = {at I t C R} be a 1-parameter subgroup of hyperbolic translations such that G = KA+K is a Cartan decomposition. Denote by ot the probability measure on G given by ot = mK * 6at * MK, where MK is the normalized Haar measure on K. Let It denote the probablity measure with density vol(Bt)>1XB,(g), where Bt = {g: d(gK, K) < t} and d denotes the canonical G-invariant Riemannian metric on G/K. Let (X, B, A) be a standard Borel probability space on which G acts measurably and ergodically by measure preserving transformations. Let 7r: G -* Iso(LP(X)) be the associated isometric representation, and let 7r(ut) and 7r(!3t) denote the operators canonically associated with Ct and It. Let n(G) = dimR G/K. We prove maximal and pointwise ergodic theorems for the ball averages It on simple real-rank one Lie groups which are analogous to N. Wiener's maximal and pointwise ergodic theorems for ball averages on in. (The general case of ball averages on semisimple groups is completed in [M-N-S].) Our main result is the following:

The first eigenvalues of finite Riemannian covers

There exists a Riemannian metric on the real projective space such that the first eigenvalue coincides with that of its Riemannian universal cover, if the dimension is bigger than 2. For the proof, we deform the canonical metric on the real projective space. A similar result is obtained for lens spaces, as well as for closed Riemannian manifolds with Riemannian double covers. As a result, on a non-orientable closed manifold other than the real projective plane, there exists a Riemannian metric such that the first eigenvalue coincides with that of its Riemannian double cover.