26 research outputs found
Fredholm vector fields and a transversality theorem
AbstractWe define the notion of a Fredholm vector field and prove a transversality result giving conditions under which a vertical family of such vector fields generically have nondegenerate zeros. Many geometric objects like minimal surfaces, geodesics, and harmonic maps arise as the zeros of a Fredholm vector field
Constrained Willmore Surfaces
Constrained Willmore surfaces are conformal immersions of Riemann surfaces
that are critical points of the Willmore energy under compactly
supported infinitesimal conformal variations. Examples include all constant
mean curvature surfaces in space forms. In this paper we investigate more
generally the critical points of arbitrary geometric functionals on the space
of immersions under the constraint that the admissible variations
infinitesimally preserve the conformal structure. Besides constrained Willmore
surfaces we discuss in some detail examples of constrained minimal and volume
critical surfaces, the critical points of the area and enclosed volume
functional under the conformal constraint.Comment: 17 pages, 8 figures; v2: Hopf tori added as an example, minor changes
in presentation, numbering changed; v3: new abstract and appendix, several
changes in presentatio
Remarks on symplectic twistor spaces
We consider some classical fibre bundles furnished with almost complex
structures of twistor type, deduce their integrability in some cases and study
\textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With
these we define a moduli space of -compatible complex structures. We
recall the theory of flag manifolds in order to study the Siegel domain and
other domains alike, which is the fibre of the referred twistor space. Finally
the structure equations for the twistor of a Riemann surface with the canonical
symplectic-metric connection are deduced, based on a given conformal coordinate
on the surface. We then relate with the moduli space defined previously.Comment: 20 pages, title changed since v2, accepted in AMPA toda
Positivity of relative canonical bundles and applications
Given a family of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle . We use a global elliptic
equation to show that this metric is strictly positive on , unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space of canonically polarized varieties
follows.
The direct images , , carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of .Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in
Invent. mat