https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on S2. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville’s equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include “solutions” of Liouville’s equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of “timelike” Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on S2 can be computed by the original Liouville path integral evaluated on a new integration cycle.

Analytic continuation of Liouville theory

A large range of sophisticated brain image analysis tools have been developed by the neuroscience community, greatly advancing the field of human brain mapping. Here we introduce the Computational Anatomy Toolbox (CAT) - a powerful suite of tools for morphometric analyses with an intuitive graphical user interface, but also usable as a shell script. CAT is suitable for beginners, casual users, experts, and developers alike providing a comprehensive set of analysis options, workflows, and integrated pipelines. The available analysis streams – illustrated on an example dataset – allow for voxel-based, surface-based, as well as region-based morphometric analyses. Importantly, CAT includes various quality control options and covers the entire analysis workflow, from cross-sectional or longitudinal data processing, to the statistical analysis, and visualization of results. The overarching aim of this article is to provide a complete description of CAT, while, at the same time, offering a citable standard reference.

CAT – A Computational Anatomy Toolbox for the Analysis of Structural MRI Data

We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg group Nil3, the universal cover of the Lie group PSL2(R) and the product spaces S2×R and H2×R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds

https://www.ems-ph.org/journals/show_pdf.php?issn=0010-2571&vol=82&iss=1&rank=4

Isometric immersions into 3-dimensional homogeneous manifolds

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

/pdf/singularities-of-flat-fronts-in-hyperbolic-space-4jhu5fkxv2.pdf

Singularities of flat fronts in hyperbolic space

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails

http://annals.math.princeton.edu/wp-content/uploads/annals-v169-n2-p03.pdf

The geometry of fronts

In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant [Br] showed that an analogue of the Weierstrass-representation formula holds for surfaces of constant mean curvature-i in the hyperbolic 3-space X3. In this article we abbreviate the term "constant mean curvature-i" as CMC-1. Like minimal surfaces in the euclidean space, the hyperbolic Gauss map of CMC-1 surfaces is defined as a holomorphic map to C U {oo}. However, in contrast to the euclidean case, the hyperbolic Gauss map of a CMC-1 surface may not be extended across the ends even if the total Gaussian curvature is finite. We call a complete CMC-1 surface, whose Gauss map can be extended across all of its ends, a CMC-1 surface of regular ends. In this article we produce an explicit tool to construct CMC-1 surfaces of regular ends. In Section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity. In our CMC-1 category, Ossermann's inequality is not expected and the Cohn-Vossen inequality is the best possible one. We show in Section 4 that the equality of the Cohn-Vossen inequality never holds for complete CMC-1 surfaces in XH3. In Section 5 we give a necessary and sufficient condition that a regular end of a CMC-1 surface be embedded. In Section 6 we classify complete CMC-1 surfaces of genus 0 with two regular ends. Our classification contains new examples. Furthermore, in Section 7, we construct several new CMC-1 surfaces with regular embedded ends. Each of these examples has a nontrivial deformation, which is mentioned in Section 3. It should be remarked that our construction does not work for surfaces with irregular ends. But there is another construction: By perturbing minimal surfaces in the euclidean 3-space, the authors constructed CMC-1 surfaces, all of whose ends are irregular (see [UY1]).

Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space

We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.

Singularities of maximal surfaces

We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show that they satisfy an Osserman-type inequality.

/pdf/maximal-surfaces-with-singularities-in-minkowski-space-5877b3iss2.pdf

Kotaro Yamada

Papers

Singularities of flat fronts in hyperbolic space

The geometry of fronts

Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space

Singularities of maximal surfaces

Maximal surfaces with singularities in Minkowski space