A thermodynamical formalism for Monge–Ampère equations,Moser–Trudinger inequalities and Kähler–Einstein metrics

We develop a variational calculus for a certain free energy functional on the space of all probability measures on a Kähler manifold X. This functional can be seen as a generalization of Mabuchi?s K-energy functional and its twisted versions to more singular situations. Applications to Monge–Ampère equations of mean field type, twisted Kähler–Einstein metrics and Moser–Trudinger type inequalities on Kähler manifolds are given. Tian?s α-invariant is generalized to singular measures, allowing in particular a proof of the existence of Kähler–Einstein metrics with positive Ricci curvature that are singular along a given anti-canonical divisor (which combined with very recent developments concerning Kähler metrics with conical singularities confirms a recent conjecture of Donaldson). As another application we show that if the Calabi flow in the (anti-)canonical class exists for all times then it converges to a Kähler–Einstein metric, when a unique one exists, which is in line with a well-known conjecture.     

On the K-stability of complete intersections in polarized manifolds

We consider the problem of existence of constant scalar curvature Kähler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai–Umemura–Tian like example of Fano 5-fold admitting no Kähler–Einstein metric, and a strong evidence of K-stability of complete intersections in Grassmannians.     

Lower bounds on the modified K-energy and complex deformations

Let (X,L)$(X,L)$ be a polarized Kähler manifold that admits an extremal metric in _c1(L)$c_1(L)$. We show that on a nearby polarized deformation (X^′,L^′)$(X^{\prime },L^{\prime })$ that preserves the symmetry induced by the extremal vector field of (X,L)$(X,L)$, the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti ,  and  to extremal metrics. Our proof also extends a convexity inequality on the space of Kähler potentials due to X.X. Chen [7] to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of CP¹×CP¹$\mathbb{C}{\mathbb{P}}^1\times \mathbb{C}{\mathbb{P}}^1$ that carry no extremal metric but with modified K-energy bounded from below.     

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g)$(M,g)$ be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.     

Extremal metrics and K-stability

We propose an algebraic geometric stability criterion for apolarised variety to admit an extremal Kähler metric. Thisgeneralises conjectures by Yau, Tian and Donaldson, which relateto the case of Kähler–Einstein and constant scalarcurvature metrics. We give a result in geometric invariant theorythat motivates this conjecture, and an example computation thatsupports it.     

Autodual Einstein Versus Kähler–Einstein

Any strictly pseudoconvex domain in carries a complete Kähler-Einstein metric, the Cheng–Yau metric, with “conformal infinity” the CR structure of the boundary.It is well known that not all CR structures on S³ arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kähler–Einstein metrics).Received: March 2004 Revision: July 2004 Accepted: August 2004     

Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications

In this paper we extend Mizoguchi–Takahashi's fixed point theorem for multi-valued mappings on a metric space endowed with a graph. As an application, we establish a fixed point theorem on an ε  -chainable metric space for mappings satisfying Mizoguchi–Takahashi contractive condition uniformly locally. Also, we establish a result on the convergence of successive approximations for certain operators (not necessarily linear) on a Banach space as another application. Consequently, this result yields the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space C[0,1]$C[0,1]$ and also enables us study the asymptotic behaviour of iterates of some nonlinear Bernstein type operators on C[0,1]$C[0,1]$.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Compactness Theorems of Extremal-Kähler Manifolds with Positive First Chern Class

In this article, we propose an extension of the compactness property for Kähler–Einstein metrics to extremal-Kähler metrics on compact Kähler manifolds with positive first Chern class and admitting non-zero holomorphic vector fields.     

Extremal Kähler metrics and complex deformation theory

Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [] has an open neighborhood inH ^1,1 (M, ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [] is nondegenerate, every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.The first author is supported in part by NSF grant DMS 92-04093.     

A trust branching path heuristic for zero–one programming

For 0–1 problems, we propose an exact Branch and Bound procedure where branching strategy is based on empirical distribution of each variable within three intervals [0,?],[?,1-?],[1-?,1]$[0\text{,}?]\text{,}[?\text{,}1-?]\text{,}[1-?\text{,}1]$ under the linear relaxation model. We compare the strategy on multiknapsack and maximum clique problems with other heuristics.     

On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)$\Phi (t):[0,\infty )\to [0,\infty )$ is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.     

Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R³,g)$({\mathbb{R}}^3,\mathfrak{g})$, where the metric g$\mathfrak{g}$ is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|²)^−ρ/2${(1+{|x|}^2)}^{-\rho /2}$ with ρ>1$\rho >1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A.     

Inverse curvature flows in asymptotically Robertson Walker spaces

In this paper we consider inverse curvature flows in a Lorentzian manifold N which is the topological product of the real numbers with a closed Riemannian manifold and equipped with a Lorentzian metric having a future singularity so that N is asymptotically Robertson Walker. The flow speeds are future directed and given by $1/F$ where F is a homogeneous degree one curvature function of class $(K^{?})$ of the principal curvatures, i.e. the n-th root of the Gauss curvature. We prove longtime existence of these flows and that the flow hypersurfaces converge to smooth functions when they are rescaled with a proper factor which results from the asymptotics of the metric.