869 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
Bimetric Theory of Fractional Quantum Hall States
We present a bimetric low-energy effective theory of fractional quantum Hall
(FQH) states that describes the topological properties and a gapped collective
excitation, known as Girvin-Macdonald-Platzman (GMP) mode. The theory consist
of a topological Chern-Simons action, coupled to a symmetric rank two tensor,
and an action \`a la bimetric gravity, describing the gapped dynamics of the
spin- GMP mode. The theory is formulated in curved ambient space and is
spatially covariant, which allows to restrict the form of the effective action
and the values of phenomenological coefficients. Using the bimetric theory we
calculate the projected static structure factor up to the order in the
momentum expansion. To provide further support for the theory, we derive the
long wave limit of the GMP algebra, the dispersion relation of the GMP mode,
and the Hall viscosity of FQH states. We also comment on the possible
applications to fractional Chern insulators, where closely related structures
arise. Finally, it is shown that the familiar FQH observables acquire a curious
geometric interpretation within the bimetric formalism.Comment: 14 pages, v2: Acknowledgments updated, v3: A few presentation
improvements, Published versio
Barcode Embeddings for Metric Graphs
Stable topological invariants are a cornerstone of persistence theory and
applied topology, but their discriminative properties are often
poorly-understood. In this paper we study a rich homology-based invariant first
defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in
the barcode space. We prove that this invariant is locally injective on the
space of metric graphs and globally injective on a GH-dense subset. Moreover,
we show that is globally injective on a full measure subset of metric graphs,
in the appropriate sense.Comment: The newest draft clarifies the proofs in Sections 7 and 8, and
provides improved figures therein. It also includes a results section in the
introductio
Patterson-Sullivan theory for Anosov subgroups
We extend several notions and results from the classical Patterson-Sullivan
theory to the setting of Anosov subgroups of higher rank semisimple Lie groups,
working primarily with invariant Finsler metrics on associated symmetric
spaces. In particular, we prove the equality between the Hausdorff dimensions
of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on
the flag manifold, and the Finsler critical exponents of Anosov subgroups.Comment: 46 pages. A gap in the proof of Theorem 8.3 in the earlier version
has been fixed. Few other minor corrections mad
Nonpositive curvature and complex analysis
We discuss a few of the metrics that are used in complex analysis and
potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic
metrics. An important feature of these metrics is that they are quite often
negatively curved. We discuss what this means and when it occurs, and proceed to
investigate some notions of nonpositive curvature, beginning with constant negative
curvature (e.g. the unit disk with the Poincaré metric), and moving on to CAT(k) and
Gromov hyperbolic spaces. We pay special attention to notions of the boundary at
infinity
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