Hawking’s Local Rigidity Theorem Without Analyticity

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop.B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.     

Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

We argue that the Einstein gravity theory can be reformulated in almost Kähler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation on manifolds enabled with nonholonomic distributions is considered. We prove that, for certain types of nonholonomic constraints, there are modelled effective Lagrangians which do not develop instabilities. It is also elaborated a linearization formalism for anholonomic noncommutative gravity theories models and analyzed the stability of stationary ellipsoidal solutions defining some nonholonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how to construct nonholonomic distributions which remove instabilities in nonsymmetric gravity theories. It is concluded that instabilities do not consist a general feature of theories of gravity with nonsymmetric metrics but a particular property of some models and/or unconstrained solutions.     

Nonholonomic Ricci Flows, Exact Solutions in Gravity, and?Symmetric and Nonsymmetric Metrics

We provide a proof that nonholonomically constrained Ricci flows of (pseudo) Riemannian metrics positively result into nonsymmetric metrics (as explicit examples, we consider flows of some physically valuable exact solutions in general relativity). There are constructed and analyzed three classes of solutions of Ricci flow evolution equations defining nonholonomic deformations of Taub NUT, Schwarzschild, solitonic and pp-wave symmetric metrics into nonsymmetric ones.     

On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy

The purpose of this note is to give a new proof of uniqueness of the Gross-Pitaevskii hierarchy, first established in [1], in a different space, based on space-time estimates similar in spirit to those of [2]. The first author is supported by the NSF grant DMS-0601186.     

Deuterium structural effects in inorganic and bioinorganic aggregates

Deuterium kinetic isotope effects are widely used in chemical and biological research. Deuterium thermodynamic effects on the aqueous synthesis of inorganic materials, however, seem not to have been recognized. Here we report that the simple replacement of H(2)O with D(2)O in the synthesis of a solid-state manganese complex results in a new structurally and magnetically distinct phase. When iron oxides are synthesized, the relative amount of the mineral phases obtained in H(2)O vs D(2)O is different. The morphology and magnetic properties of the iron core of the iron storage protein ferritin are likewise different when mineralization is carried out in heavy water. The formation of extra inorganic solids, change in the ratio of two phases or alteration of a single phase morphology in D(2)O suggest that new inorganic and bioinorganic metal complexes might be obtained by using the thermodynamic isotope effect.     

Adiabatic limits of eta and zeta functions of elliptic operators

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator , constructed from an elliptic family of operators indexed by S ¹ . We show that the regularized values ( _t ,0) and t( _t ,0) are smooth functions of t at t=0, and we identify their values at t=0 with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ( _t ,s) and t( _t ,s) are shown to extend smoothly to t=0 for all values of s. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms. Mathamatics Subject Classification (2000): 58J28, 58J52Partially supported by ANSTI (Romania), the European Commission RTN HPRN-CT-1999-00118 Geometric Analysis and by the IREX RTR project.     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.     

On a Class of Deformations of Compressible, Isotropic, Nonlinearly Elastic Solids

We consider deformations of unconstrained, isotropic hyperelastic solids which satisfy the condition that the determinant of the deformation gradient is constant. In the absence of body forces, it is shown (i) that a certain deformation in this class (which describes the bending of rectangular blocks into annular cylindrical sectors) is not possible in any of the considered materials, (ii) that in the case when the body fills the whole space, it is composed of a compressible neo-Hookean material and it is subjected to relatively moderate loads, these deformations are necessarily homogeneous and (iii) that for boundary conditions of place and relative to a certain sub-class of the class of considered materials, these deformations are globally stable, in the sense that they are minimizers for the total energy with respect to smooth variations that are compatible with the boundary conditions. This revised version was published online in August 2006 with corrections to the Cover Date.