14,533 research outputs found
Discrete Minimal Surface Algebras
We consider discrete minimal surface algebras (DMSA) as generalized
noncommutative analogues of minimal surfaces in higher dimensional spheres.
These algebras appear naturally in membrane theory, where sequences of their
representations are used as a regularization. After showing that the defining
relations of the algebra are consistent, and that one can compute a basis of
the enveloping algebra, we give several explicit examples of DMSAs in terms of
subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by
itself). A special class of DMSAs are Yang-Mills algebras. The representation
graph is introduced to study representations of DMSAs of dimension d<=4, and
properties of representations are related to properties of graphs. The
representation graph of a tensor product is (generically) the Cartesian product
of the corresponding graphs. We provide explicit examples of irreducible
representations and, for coinciding eigenvalues, classify all the unitary
representations of the corresponding algebras
Lectures on Minimal Surface Theory
An article based on a four-lecture introductory minicourse on minimal surface
theory given at the 2013 summer program of the Institute for Advanced Study and
the Park City Mathematics Institute.Comment: 46 pages, 6 figures. Some references added/corrected on August 2,
2014. A few minor corrections on October 16, 2015. Additional typos corrected
on January 17, 201
A minimal surface with unbounded curvature
We construct a complete, embedded minimal surface in euclidean 3-space which
has unbounded Gaussian curvature. It has infinite genus, infinitely many
catenoidal type ends and one limit end.Comment: 24 pages, 2 figure
Minimal surface singularities are Lipschitz normally embedded
Any germ of a complex analytic space is equipped with two natural metrics:
the {\it outer metric} induced by the hermitian metric of the ambient space and
the {\it inner metric}, which is the associated riemannian metric on the germ.
We show that minimal surface singularities are Lipschitz normally embedded
(LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and
inner metrics, and that they are the only rational surface singularities with
this property.Comment: This paper is a major revision of the 2015 version. It now builds on
the paper arXiv:1806.11240 by the same authors which gives a general
characterization of Lipschitz normally embedded surface singularitie
Lawson's genus two minimal surface and meromorphic connections
We investigate the Lawson genus surface by methods from integrable system
theory. We prove that the associated family of flat connections comes from a
family of flat connections on a punctured sphere. We describe the
symmetries of the holonomy and show that it is already determined by the
holonomy around one of the punctures. We show the existence of a meromorphic
DPW potential for the Lawson surface which is globally defined on the surface.
We determine this potential explicitly up to two unknown functions depending
only on the spectral parameter
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