Search results for "Differential geometry"
showing 10 items of 462 documents
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
2016
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Geodesics on spaces of almost hermitian structures
1994
A natural metric on the space of all almost hermitian structures on a given manifold is investigated.
A note on Sobolev isometric immersions below W2,2 regularity
2017
Abstract This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W 2 , 2 setting. We show that the Hessian of each coordinate function of a W 2 , p , p 2 , isometric immersion satisfies a low rank property in the almost everywhere sense, in particular, its Gaussian curvature vanishes almost everywhere. Meanwhile, we provide an example of a W 2 , p , p 2 , isometric immersion from a bounded domain of R 2 into R 3 that has multiple singularities.
Magnetised Polish doughnuts revisited
2017
We discuss a procedure to build new sequences of magnetised, equilibrium tori around Kerr black holes which combines two approaches previously considered in the literature. For simplicity we assume that the test-fluid approximation holds, and hence we neglect the self-gravity of the fluid. The models are built assuming a particular form of the angular momentum distribution from which the location and morphology of equipotential surfaces can be computed. This ansatz includes, in particular, the constant angular momentum case originally employed in the construction of thick tori - or Polish doughnuts - and it has already been used to build equilibrium sequences of purely hydrodynamical models…
Caustics for spherical waves
2016
We study the development of caustics in shift-symmetric scalar field theories by focusing on simple waves with an $SO(p)$-symmetry in an arbitrary number of space dimensions. We show that the pure Galileon, the DBI-Galileon, and the extreme-relativistic Galileon naturally emerge as the unique set of caustic-free theories, highlighting a link between the caustic-free condition for simple $SO(p)$-waves and the existence of either a global Galilean symmetry or a global (extreme-)relativistic Galilean symmetry.
The 1-loop effective potential for the Standard Model in curved spacetime
2018
The renormalisation group improved Standard Model effective potential in an arbitrary curved spacetime is computed to one loop order in perturbation theory. The loop corrections are computed in the ultraviolet limit, which makes them independent of the choice of the vacuum state and allows the derivation of the complete set of $\beta$-functions. The potential depends on the spacetime curvature through the direct non-minimal Higgs-curvature coupling, curvature contributions to the loop diagrams, and through the curvature dependence of the renormalisation scale. Together, these lead to significant curvature dependence, which needs to be taken into account in cosmological applications, which i…
A construction of Frobenius manifolds from stability conditions
2018
A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to each mutation of $Q$ and evaluating the families at a special point yields a different branch of the m…
The Schouten - Nijenhuis bracket, cohomology and generalized Poisson structures
1996
Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.
Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians
1997
In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians $G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational quantum c…
Acceleration radiation, transition probabilities, and trans-Planckian physics
2010
An important question in the derivation of the acceleration radiation, which also arises in Hawking's derivation of black hole radiance, is the need to invoke trans-Planckian physics in describing the creation of quanta. We point out that this issue can be further clarified by reconsidering the analysis in terms of particle detectors, transition probabilities and local two-point functions. By writing down separate expressions for the spontaneous-and induced-transition probabilities of a uniformly accelerated detector, we show that the bulk of the effect comes from the natural (non-trans-Planckian) scale of the problem, which largely diminishes the importance of the trans-Planckian sector. T…