2,986 research outputs found
Infinitely many solutions to the Yamabe problem on noncompact manifolds
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, , , , and , . As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on , for all , the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in are periods of bifurcating branches of solutions to the Yamabe problem on , ,
Genericity of nondegenerate geodesics with general boundary conditions
Let M be a possibly noncompact manifold. We prove, generically in the
C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on
M do not possess any degenerate geodesics satisfying suitable boundary
conditions. This extends a result of Biliotti, Javaloyes and Piccione for
geodesics with fixed endpoints to the case where endpoints lie on a compact
submanifold P of the product MxM that satisfies an admissibility condition.
Such condition holds, for example, when P is transversal to the diagonal of
MxM. Further aspects of these boundary conditions are discussed and general
conditions under which metrics without degenerate geodesics are C^k-generic are
given.Comment: LaTeX2e, 21 pages, no figure
Teichmüller theory and collapse of flat manifolds
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified
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