Parabolic equations of Von Karman type on Kähler manifolds, II

We complete the study of a parabolic version of a system of Von Karman type on a compact Kähler manifold of complex dimension m. We consider a family of problems k(P). We prove existence of local in time solutions when k=0. When −m?k<0, we define a notion of weak solution, and give some uniqueness and existence results.     

Indefinite locally conformal Kähler manifolds

We study the basic properties of an indefinite locally conformal Kähler (l.c.K.) manifold. Any indefinite l.c.K. manifold M with a parallel Lee form ω is shown to possess two canonical foliations F and Fc, the first of which is given by the Pfaff equation ω=0 and the second is spanned by the Lee and the anti-Lee vectors of M. We build an indefinite l.c.K. metric on the noncompact complex manifold Ω₊=(Λ₊?Λ₀)/Gλ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of F when M=Ω₊ (where is −²|z₁|−?−²|zs|+²|zs+1|+?+²|zn|>0). We study the geometry of the second fundamental form of the leaves of F and Fc. In the degenerate cases (corresponding to a lightlike Lee vector) we use the technique of screen distributions and (lightlike) transversal bundles developed by A. Bejancu et al. [K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dordrecht, 1996].     

Nearly Kähler manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

We study the local structures of nearly Kähler manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke (TV). We show that there does not exist a TV Bochner flat strict nearly Kähler manifold in 2n(?10) dimension and determine the local structures of the manifolds in 6 and 8 dimensions.     

Kähler non-collapsing,eigenvalues and the Calabi flow

We first prove a new compactness theorem of Kähler metrics, which confirms a prediction in [17]. Then we establish several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed for the Kähler–Ricci flow in [22] to obtain several new small energy theorems of the Calabi flow.     

On the spectrum of the twisted Dolbeault Laplacian over Kähler manifolds

We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact Kähler manifolds.     

Stratified Kähler structures on adjoint quotients

Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kähler structure having twice the kinetic energy of the metric as its potential, and Kähler reduction with reference to the adjoint action yields a stratified Kähler structure on the resulting adjoint quotient. Exploiting classical invariant theory, in particular bisymmetric functions and variants thereof, we explore the singular Poisson-Kähler geometry of this quotient. Among other things we prove that, for various compact groups, the real coordinate ring of the adjoint quotient is generated, as a Poisson algebra, by the real and imaginary parts of the fundamental characters. We also show that singular Kähler quantization of the geodesic flow on the reduced level yields the irreducible algebraic characters of the complexified group.     

Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics

We propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kähler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kähler and conformal geometry such as constant scalar curvature metrics, Kähler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kähler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere.     

On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds

We obtain a stability estimate for the degenerate complex Monge-Ampère operator which generalizes a result of Ko?odziej (2003) [12]. In particular, we obtain the optimal stability exponent and also treat the case when the right-hand side is a general Borel measure satisfying certain regularity conditions. Moreover, our result holds for functions plurisubharmonic with respect to a big form, thus generalizing the Kähler form setting in Ko?odziej (2003) [12]. Independently, we also provide more detail for the proof in Zhang (2006) [18] on continuity of the solution with respect to a special big form when the right-hand side is Lp-measure with p>1.     

Holomorphic cohomology groups on compact Kähler complex spaces

Let (X,OX) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups Hq(X,C) carry a mixed Hodge structure. In particular they carry a weight filtration W−lHq(X,C) (l=0,…,q), and the graded quotients have a direct sum decomposition into holomorphic invariants as . Here we investigate the relationships between the above invariants for r=0 and the cohomology groups , where is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c₁(L) in has pure type (1,1).     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Volume renormalization for complete Einstein-Kähler metrics

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern-Gauss-Bonnet formula. Relations between renormalized volume and CR Q-curvature are also investigated.     

A Laplace integral on a Kähler manifold and Calabi's diastasis function

In this paper we give a different proof of Engliš's result [J. Reine Angew. Math. 528 (2000) 1-39] about the asymptotic expansion of a Laplace integral on a real analytic Kähler manifold (M,g) by using the link between the metric g and the associated Calabi's diastasis function D. We also make explicit the connection between the coefficients of Engliš' expansion and Gray's invariants [Michigan Math. J. (1973) 329-344].     

Algebraic deformations of compact Kähler surfaces

A short proof is given of Kodaira's result that every compact kähler surface is a deformation of an algebraic surface under the extra assumption that all infinitesimal deformations are unobstructed.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

Conformally Kähler base metrics for Einstein warped products

A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP¹ bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (Md,g), where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.