A note on inverse curvature flows in ARW spacetimes
We prove that the leaves of the rescaled curvature flow considered in arXiv:math/0403485 [math.DG] converge to the graph of a constant function.
math.DG↗
arXiv subjects
Claus Gerhardt
Publications and source records attributed to Claus Gerhardt.
A unified quantum theory II: gravity interacting with Yang-Mills and spinor fields
We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation in a vector bundle and the method of second quantization leads to a symplectic vector space $(V,\om)$ and a corresponding CCR representation for the bosonic components and a CAR representation for the fermionic part. The solution space of the Wheeler-DeWitt equation is invariant under gauge transformations and under isometries in the spacelike base space $\so$ of a given Riemannian metric $ρ_{ij}$. We also define a net of local subalgebras which satisfy four of the Haag-Kastler axioms.
math-ph↗
A unified quantum theory I: gravity interacting with a Yang-Mills field
Using the results and techniques of a previous paper where we proved the quantization of gravity we extend the former result by adding a Yang-Mills functional and a Higgs term to the Einstein-Hilbert action.
hep-th↗
The quantization of gravity in globally hyperbolic spacetimes
We apply the ADM approach to obtain a Hamiltonian description of the Einstein-Hilbert action. In doing so we add four new ingredients: (i) We eliminate the diffeomorphism constraints. (ii) We replace the densities $\sqrt g$ by a function $\f(x,g_{ij})$ with the help of a fixed metric $χ$ such that the Lagrangian and hence the Hamiltonian are functions. (iii) We consider the Lagrangian to be defined in a fiber bundle with base space $\so$ and fibers F(x) which can be treated as Lorentzian manifolds equipped with the Wheeler-DeWitt metric. It turns out that the fibers are globally hyperbolic. (iv) The Hamiltonian operator $H$ is a normally hyperbolic operator in the bundle acting only in the fibers and the Wheeler-DeWitt equation $Hu=0$ is a hyperbolic equation in the bundle. Since the corresponding Cauchy problem can be solved for arbitrary smooth data with compact support, we then apply the standard techniques of QFT which can be naturally modified to work in the bundle.
gr-qc↗
Non-scale-invariant inverse curvature flows in Euclidean space
We consider the inverse curvature flows $\dot x=F^{-p}ν$ of closed star-shaped hypersurfaces in Euclidean space in case $0 1$, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to the unit sphere.
math.DG↗
Combining gravity with the forces of the standard model on a cosmological scale
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the underlying spacetime is a Friedman universe with flat spatial slices and where the matter fields are comprised of the strong interaction, with $\SU(3)$ replaced by a general $\SU(n)$, $n\ge 2$, and the electro-weak interaction. The wave functions are maps from $\R[4n+10]$ to a subspace of the antisymmetric Fock space, and one noteworthy result is that, whenever the electro-weak interaction is involved, the image of an eigenfunction is in general not one dimensional, i.e., in general it makes no sense specifying a fermion and looking for an eigenfunction the range of which is contained in the one dimensional vector space spanned by the fermion.
gr-qc↗
Quantum cosmological Friedman models with a Yang-Mills field and positive energy levels
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a Yang-Mills field, with or without mass term, if the spatial geometry of the underlying spacetime is homothetic to $\R[3]$. The energy levels of the resulting quantum model, i.e., the eigenvalues of the corresponding self-adjoint Hamiltonian with a pure point spectrum, are strictly positive.
gr-qc↗
An energy gap for Yang-Mills connections
Consider a Yang-Mills connection over a Riemann manifold $M=M^n$, $n\ge 3$, where $M$ may be compact or complete. Then its energy must be bounded from below by some positive constant, if $M$ satisfies certain conditions, unless the connection is flat.
math.DG↗
A general existence proof for non-linear elliptic equations in semi-Riemannian spaces
We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation.
math.DG↗
Quantum cosmological Friedman models with a massive Yang-Mills field
We prove the existence of a spectral resolution of the Wheeler-DeWitt equation when the matter field is provided by a massive Yang-Mills field. The resolution is achieved by first solving the free eigenvalue problem for the gravitational field and then the constrained eigenvalue problem for the Yang-Mills field. In the latter case the mass of the Yang-Mills field assumes the role of the eigenvalue.
gr-qc↗
Quantum cosmological Friedman models with an initial singularity
We consider the Wheeler-DeWitt equation $Hψ=0$ in a suitable Hilbert space. It turns out that this equation has countably many solutions $ψ_i$ which can be considered as eigenfunctions of a Hamilton operator implicitly defined by $H$. We consider two models, a bounded one, $0<r<r_0$, and an unbounded, $0<r<\un$, which represent different eigenvalue problems. In the bounded model we look for eigenvalues $\Lam_i$, where the $\Lam_i$ are the values of the cosmological constant which we used in the Einstein-Hilbert functional, and in the unbounded model the eigenvalues are given by $(-\Lam_i)^{-\frac {n-1}{n}}$, where $\Lam_i<0$. The $ψ_i$ form a basis of the underlying Hilbert space. All solutions have an initial singularity in $r=0$. Under certain circumstances a smooth transition from big crunch to big bang is possible.
gr-qc↗
The inverse mean curvature flow in cosmological spacetimes
We prove that the leaves of an inverse mean curvature flow provide a foliation of a future end of a cosmological spacetime $N$ under the necessary and sufficent assumptions that $N$ satisfies a future mean curvature barrier condition and a strong volume decay condition. Moreover, the flow parameter $t$ can be used to define a new physically important time function.
math.DG↗
Curvature flows and CMC hypersurfaces
We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.
math.DG↗
Curvature flows in semi-Riemannian manifolds
We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.
math.DG↗
Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds
We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class $C^{m,\al}$, $m\ge4$.
math.DG↗
Minkowski type problems for convex hypersurfaces in the sphere
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.
math.DG↗
On the CMC foliation of future ends of a spacetime
We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volume of any sequence of spacelike hypersurfaces, which run into the future singularity, decays to zero provided there exists a time function covering a future end, such that the level hypersurfaces have non-negative mean curvature and decaying volume.
math.DG↗
Branes, moduli spaces and smooth transition from big crunch to big bang
We consider branes $N$ in a Schwarzschild-$\text{AdS}_{(n+2)}$ bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential $V$, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field equation that is smooth across the singularity $r=0$, and which has smooth and uniquely determined solutions which exist across the singularity in an interval $(-\e,\e)$. Restricting a solution to $(-\e,0)$ \resp $(0,\e)$, and assuming $n$ odd, we obtain branes $N$ \resp $\hat N$ which together form a smooth hypersurface. Thus a smooth transition from big crunch to big bang is possible both geometrically as well as physically.
hep-th↗