24,748 research outputs found
The homogeneity theorem for supergravity backgrounds
We prove the strong homogeneity conjecture for eleven- and ten-dimensional
(Poincar\'e) supergravity backgrounds. In other words, we show that any
backgrounds of 11-dimensional, type I/heterotic or type II supergravity
theories preserving a fraction greater than one half of the supersymmetry of
the underlying theory are necessarily locally homogeneous. Moreover we show
that the homogeneity is due precisely to the supersymmetry, so that at every
point of the spacetime one can find a frame for the tangent space made out of
Killing vectors constructed out of the Killing spinors.Comment: 8 page
Homogeneous matchbox manifolds
We prove that a homogeneous matchbox manifold of any finite dimension is
homeomorphic to a McCord solenoid, thereby proving a strong version of a
conjecture of Fokkink and Oversteegen. The proof uses techniques from the
theory of foliations that involve making important connections between
homogeneity and equicontinuity. The results provide a framework for the study
of equicontinuous minimal sets of foliations that have the structure of a
matchbox manifold.Comment: This is a major revision of the original article. Theorem 1.4 has
been broadened, in that the assumption of no holonomy is no longer required,
only that the holonomy action is equicontinuous. Appendices A and B have been
removed, and the fundamental results from these Appendices are now contained
in the preprint, arXiv:1107.1910v
On the formal structure of logarithmic vector fields
In this article, we prove that a free divisor in a three dimensional complex
manifold must be Euler homogeneous in a strong sense if the cohomology of its
complement is the hypercohomology of its logarithmic differential forms. F.J.
Calderon-Moreno et al. conjectured this implication in all dimensions and
proved it in dimension two. We prove a theorem which describes in all
dimensions a special minimal system of generators for the module of formal
logarithmic vector fields. This formal structure theorem is closely related to
the formal decomposition of a vector field by Kyoji Saito and is used in the
proof of the above result. Another consequence of the formal structure theorem
is that the truncated Lie algebras of logarithmic vector fields up to dimension
three are solvable. We give an example that this may fail in higher dimensions.Comment: 13 page
Locally -homogeneous Busemann -spaces
We present short proofs of all known topological properties of general
Busemann -spaces (at present no other property is known for dimensions more
than four). We prove that all small metric spheres in locally -homogeneous
Busemann -spaces are homeomorphic and strongly topologically homogeneous.
This is a key result in the context of the classical Busemann conjecture
concerning the characterization of topological manifolds, which asserts that
every -dimensional Busemann -space is a topological -manifold. We also
prove that every Busemann -space which is uniformly locally -homogeneous
on an orbal subset must be finite-dimensional
The Stokes conjecture for waves with vorticity
We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity.
We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity.
In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity
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