230 research outputs found
Hyperbolic Unfoldings of Minimal Hypersurfaces
We study the intrinsic geometry of area minimizing (and also of almost
minimizing) hypersurfaces from a new point of view by relating this subject to
quasiconformal geometry. For any such hypersurface we define and construct a
so-called S-structure which reveals some unexpected geometric and analytic
properties of the hypersurface and its singularity set. In this paper, this is
used to prove the existence of hyperbolic unfoldings: canonical conformal
deformations of the regular part of these hypersurfaces into complete Gromov
hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to the
singular set
Differential K-theory. A survey
Generalized differential cohomology theories, in particular differential
K-theory (often called "smooth K-theory"), are becoming an important tool in
differential geometry and in mathematical physics. In this survey, we describe
the developments of the recent decades in this area. In particular, we discuss
axiomatic characterizations of differential K-theory (and that these uniquely
characterize differential K-theory). We describe several explicit
constructions, based on vector bundles, on families of differential operators,
or using homotopy theory and classifying spaces. We explain the most important
properties, in particular about the multiplicative structure and push-forward
maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer
family index theorem for differential K-theory.Comment: 50 pages, report based in particular on work done sponsored the DFG
SSP "Globale Differentialgeometrie". v2: final version (only typos
corrected), to appear in C. B\"ar et al. (eds.), Global Differential
Geometry, Springer Proceedings in Mathematics 17, Springer-Verlag Berlin
Heidelberg 201
Prescribing eigenvalues of the Dirac operator
In this note we show that every compact spin manifold of dimension
can be given a Riemannian metric for which a finite part of the spectrum of the
Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity
1.Comment: To appear in Manuscripta Mathematic
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