57 research outputs found
Variational aspects of Laplace eigenvalues on Riemannian surfaces
We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of λk-extremal metrics and the existence of a partially regular λ₁-maximiser
On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces
A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the multiplicities of the eigenvalues of the Schrödinger operator.(-Δg + υ), where υ is C∞-smooth, on a compact Riemannian surface M are bounded in terms of the eigenvalue index and the genus of M. We prove that these multiplicity bounds hold for an Lp-potential υ, where p > 1. We also discuss similar multiplicity bounds for Laplace eigenvalues on singular Riemannian surfaces
Nematic Structure of Space-Time and its Topological Defects in 5D Kaluza-Klein Theory
We show, that classical Kaluza-Klein theory possesses hidden nematic
dynamics. It appears as a consequence of 1+4-decomposition procedure, involving
4D observers 1-form \lambda. After extracting of boundary terms the, so called,
"effective matter" part of 5D geometrical action becomes proportional to square
of anholonomicity 3-form \lambda\wedge d\lambda. It can be interpreted as twist
nematic elastic energy, responsible for elastic reaction of 5D space-time on
presence of anholonomic 4D submanifold, defined by \lambda. We derive both 5D
covariant and 1+4 forms of 5D nematic equilibrium equations, consider simple
examples and discuss some 4D physical aspects of generic 5D nematic topological
defects.Comment: Latex-2e, 14 pages, 1 Fig., submitted to GR
Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry
Fibrations and fundamental groups of Kaehler-Weyl manifolds
We extend the Siu--Beauville theorem to a certain class of compact
Kaehler--Weyl manifolds, proving that they fiber holomorphically over
hyperbolic Riemannian surfaces whenever they satisfy the necessary topological
hypotheses. As applications we obtain restrictions on the fundamental groups of
such Kaehler--Weyl manifolds, and show that in certain cases they are in fact
Kaehler.Comment: minor changes and addition of a postscript, now 13 pages; to appear
in Proc. Amer. Math. So
Bounds for Laplace eigenvalues of Kähler metrics
We prove inequalities for Laplace eigenvalues of Kähler manifolds generalising to higher eigenvalues the classical inequality for the first Laplace eigenvalue due to Bourguignon, Li, and Yau in 1994. We also obtain similar eigenvalue inequalities for analytic varieties in Kähler manifolds
The Effective Energy-Momentum Tensor in Kaluza-Klein Gravity With Large Extra Dimensions and Off-Diagonal Metrics
We consider a version of Kaluza-Klein theory where the cylinder condition is
not imposed. The metric is allowed to have explicit dependence on the "extra"
coordinate(s). This is the usual scenario in brane-world and space-time-matter
theories. We extend the usual discussion by considering five-dimensional
metrics with off-diagonal terms. We replace the condition of cylindricity by
the requirement that physics in four-dimensional space-time should remain
invariant under changes of coordinates in the five-dimensional bulk. This
invariance does not eliminate physical effects from the extra dimension but
separates them from spurious geometrical ones. We use the appropriate splitting
technique to construct the most general induced energy-momentum tensor,
compatible with the required invariance. It generalizes all previous results in
the literature. In addition, we find two four-vectors, J_{m}^{mu} and
J_{e}^{mu}, induced by off-diagonal metrics, that separately satisfy the usual
equation of continuity in 4D. These vectors appear as source-terms in equations
that closely resemble the ones of electromagnetism. These are Maxwell-like
equations for an antisymmetric tensor {F-hat}_{mu nu} that generalizes the
usual electromagnetic one. This generalization is not an assumption, but
follows naturally from the dimensional reduction. Thus, if {F-hat}_{mu nu}
could be identified with the electromagnetic tensor, then the theory would
predict the existence of classical magnetic charge and current. The splitting
formalism used allows us to construct 4D physical quantities from
five-dimensional ones, in a way that is independent on how we choose our
space-time coordinates from those of the bulk.Comment: New title, editorial changes made as to match the version to appear
in International Journal of Modern Physics
Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound
We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed
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