55,366 research outputs found
Holographic Holes and Differential Entropy
Recently, it has been shown by Balasubramanian et al. and Myers et al. that
the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in
the bulk of a holographic spacetime has an interpretation as the differential
entropy of a particular family of intervals (or strips) in the boundary theory.
We first extend this construction to bulk surfaces which vary in time. We then
give a general proof of the equality between the gravitational entropy and the
differential entropy. This proof applies to a broad class of holographic
backgrounds possessing a generalized planar symmetry and to certain classes of
higher-curvature theories of gravity. To apply this theorem, one can begin with
a bulk surface and determine the appropriate family of boundary intervals by
considering extremal surfaces tangent to the given surface in the bulk.
Alternatively, one can begin with a family of boundary intervals; as we show,
the differential entropy then equals the gravitational entropy of a bulk
surface that emerges from the intersection of the neighboring entanglement
wedges, in a continuum limit.Comment: 62 pages; v2: minor improvements to presentation, references adde
About the hyperbolicity of complete intersections
This note is an extended version of a thirty minutes talk given at the "XIX
Congresso dell'Unione Matematica Italiana", held in Bologna from September 12th
to September 17th, 2011. This was essentially a survey talk about connections
between Kobayashi hyperbolicity properties and positivity properties of the
canonical bundle of projective algebraic varieties.Comment: 9 pages, no figures, to appear on Boll. Unione Mat. Ita
Solitons and admissible families of rational curves in twistor spaces
It is well known that twistor constructions can be used to analyse and to
obtain solutions to a wide class of integrable systems. In this article we
express the standard twistor constructions in terms of the concept of an
admissible family of rational curves in certain twistor spaces. Examples of of
such families can be obtained as subfamilies of a simple family of rational
curves using standard operations of algebraic geometry. By examination of
several examples, we give evidence that this construction is the basis of the
construction of many of the most important solitonic and algebraic solutions to
various integrable differential equations of mathematical physics. This is
presented as evidence for a principal that, in some sense, all soliton-like
solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the
objectives of the paper. This is the final version which will appear in
Nonlinearit
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