592 research outputs found
The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2
We develop a general method to construct subsets of complete Riemannian
manifolds that cannot contain images of non-constant harmonic maps from
compact manifolds. We apply our method to the special case where the harmonic
map is the Gauss map of a minimal submanifold and the complete manifold
is a Grassmannian. With the help of a result by Allard [Allard, W. K. (1972). On the first variation of a varifold. Annals of mathematics, 417-491.], we can study
the graph case and have an approach to prove Bernstein-type theorems. This
enables us to extend Moser’s Bernstein theorem [Moser, J. (1961). On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics, 14(3), 577-591.] to codimension two, i.e., a minimal p-submanifold in , which is the graph of a smooth function defined
on the entire with bounded slope, must be a p-plane
Harmonic maps from surfaces of arbitrary genus into spheres
We relate the existence problem of harmonic maps into S2 to the convex geometry of S2. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into S2. On the other hand, we produce new examples of regions that do not contain closed geodesics (that is, harmonic maps from S1) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex functions. Our construction uses M. Struwe’s heat flow approach for the existence of harmonic maps from surfaces
Topological gauge-gravity equivalence: fiber bundle and homology aspects
In the works of A. Ach\'ucarro and P. K. Townsend and also by E. Witten, a
duality between three-dimensional Chern-Simons gauge theories and gravity was
established. First (Ach\'ucarro and Townsend), by considering an
In\"on\"u-Wigner contraction from a superconformal gauge theory to an Anti-de
Sitter supergravity. Then, Witten was able to obtain, from Chern-Simons theory
(in two cases: Poincar\'e and de Sitter gauge theories), an Einstein-Hilbert
gravity by mapping the gauge symmetry in local isometries and diffeomorphisms.
In all cases, the results made use of the field equations. Latter, we were
capable to generalize Witten's work (in Euclidean spacetime) to the off-shell
cases, as well as to four dimensional Yang-Mills theory with de Sitter gauge
symmetry. The price we paid is that curvature and torsion must obey some
constraints under the action of the interior derivative. These constraints
implied on the partial breaking of diffeomorphism invariance. In the present
work, we, first, formalize our early results in terms of fiber bundle theory by
establishing the formal aspects of the map between a principal bundle (gauge
theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism
invariance. Then, we study the effect of the constraints on the homology
defined by the interior derivative. The result being the emergence of a
nontrivial homology in Riemann-Cartan manifolds.Comment: 14 pages; 5 figure
Constrained gauge-gravity duality in three and four dimensions
The equivalence between Chern-Simons and Einstein-Hilbert actions in three
dimensions established by A.~Ach\'ucarro and P.~K.~Townsend (1986) and
E.~Witten (1988) is generalized to the off-shell case. The technique is also
generalized to the Yang-Mills action in four dimensions displaying de Sitter
gauge symmetry. It is shown that, in both cases, we can directly identify a
gravity action while the gauge symmetry can generate spacetime local isometries
as well as diffeomorphisms. The price we pay for working in an off-shell
scenario is that specific geometric constraints are needed. These constraints
can be identified with foliations of spacetime. The special case of spacelike
leafs evolving in time is studied. Finally, the whole set up is analyzed under
fiber bundle theory. In this analysis we show that a traditional gauge theory,
where the gauge field does not influence in spacetime dynamics, can be (for
specific cases) consistently mapped into a gravity theory in the first order
formalism.Comment: 25 pages. No figures. Final version accepted for publication at the
European Physical Journal
Graphical mean curvature flow with bounded bi-Ricci curvature
We consider the graphical mean curvature flow of strictly area decreasing
maps , where is a compact Riemannian manifold of dimension
and a complete Riemannian surface of bounded geometry. We prove long-time
existence of the flow and that the strictly area decreasing property is
preserved, when the bi-Ricci curvature of is bounded from below by
the sectional curvature of . In addition, we obtain smooth
convergence to a minimal map if . These
results significantly improve known results on the graphical mean curvature
flow in codimension
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