1,280 research outputs found
Generalized Lawson tori and Klein bottles
Using Takahashi theorem we propose an approach to extend known families of
minimal tori in spheres. As an example, the well-known two-parametric family of
Lawson tau-surfaces including tori and Klein bottles is extended to a
three-parametric family of tori and Klein bottles minimally immersed in
spheres. Extremal spectral properties of the metrics on these surfaces are
investigated. These metrics include i) both metrics extremal for the first
non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and
the metric on the equilateral torus and ii) the metric maximal for the first
non-trivial eigenvalue on the Klein bottle.Comment: 17 pages, v.2: minor correction
The Classification of Branched Willmore Spheres in the -Sphere and the -Sphere
We extend the classification of Robert Bryant of Willmore spheres in to
variational branched Willmore spheres and show that they are inverse
stereographic projections of complete minimal surfaces with finite total
curvature in and vanishing flux. We also obtain a classification
of variational branched Willmore spheres in , generalising a theorem of
Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points,
we obtain an improved regularity of the unit normal of variational
branched Willmore surfaces in arbitrary codimension. We also prove that the
width of Willmore sphere min-max procedures in dimension and , such as
the sphere eversion, is an integer multiple of .Comment: 74 pages, 1 figur
A new construction of homogeneous quaternionic manifolds and related geometric structures
Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a
module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic
manifold (M,Q) is constructed for any spin(V)-equivariant linear map \Pi :
\wedge^2 W \to V. If the skew symmetric vector valued bilinear form \Pi is
nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g
such that (M,Q,g) is a homogeneous quaternionic pseudo-K\"ahler manifold.
The construction is shown to have a natural mirror in the category of
supermanifolds. In fact, for any spin(V)-equivariant linear map \Pi : Sym^2 W
\to V a homogeneous quaternionic supermanifold (M,Q) is constructed and,
moreover, a homogeneous quaternionic pseudo-K\"ahler supermanifold (M,Q,g) if
the symmetric vector valued bilinear form \Pi is nondegenerate.Comment: to appear in the Memoirs of the AMS, 81 pages, Latex source fil
The Willmore flow of Hopf-tori in the -sphere
In this article the author investigates flow lines of the classical Willmore
flow, which start moving in a parametrization of a Hopf-torus in
. We prove that any such flow line of the Willmore flow exists
globally, in particular does not develop any singularities, and subconverges to
some smooth Willmore-Hopf-torus in every -norm. Moreover, if in addition
the Willmore-energy of the initial immersion is required to be smaller
than the threshold , then the unique flow line of
the Willmore flow, starting to move in , converges fully to a conformal
image of the standard Clifford-torus in every -norm, up to time
dependent, smooth reparametrizations. Key instruments for the proofs are the
equivariance of the Hopf-fibration w.r.t.
the effect of the -gradient of the Willmore energy applied to smooth
Hopf-tori in and to smooth closed regular curves in
, a particular version of the Lojasiewicz-Simon gradient
inequality, and a certain mathematical bridge between the Euler-Lagrange
equation of the elastic energy functional and a particular class of elliptic
curves over
Orientation theory in arithmetic geometry
This work is devoted to study orientation theory in arithmetic geometric
within the motivic homotopy theory of Morel and Voevodsky. The main tool is a
formulation of the absolute purity property for an \emph{arithmetic cohomology
theory}, either represented by a cartesian section of the stable homotopy
category or satisfying suitable axioms. We give many examples, formulate
conjectures and prove a useful property of analytical invariance. Within this
axiomatic, we thoroughly develop the theory of characteristic and fundamental
classes, Gysin and residue morphisms. This is used to prove Riemann-Roch
formulas, in Grothendieck style for arbitrary natural transformations of
cohomologies, and a new one for residue morphisms. They are applied to rational
motivic cohomology and \'etale rational -adic cohomology, as expected by
Grothendieck in \cite[XIV, 6.1]{SGA6}.Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference
in the Tata Institute. Thanks a lot goes to the referee for his enormous work
(more than 100 comments) which was of great help. Among these corrections, he
indicated to me a sign mistake in formula (3.2.14.a) which was very hard to
detec
On surfaces with prescribed shape operator
The problem of immersing a simply connected surface with a prescribed shape
operator is discussed. From classical and more recent work, it is known that,
aside from some special degenerate cases, such as when the shape operator can
be realized by a surface with one family of principal curves being geodesic,
the space of such realizations is a convex set in an affine space of dimension
at most 3. The cases where this maximum dimension of realizability is achieved
have been classified and it is known that there are two such families of shape
operators, one depending essentially on three arbitrary functions of one
variable (called Type I in this article) and another depending essentially on
two arbitrary functions of one variable (called Type II in this article).
In this article, these classification results are rederived, with an emphasis
on explicit computability of the space of solutions. It is shown that, for
operators of either type, their realizations by immersions can be computed by
quadrature. Moreover, explicit normal forms for each can be computed by
quadrature together with, in the case of Type I, by solving a single linear
second order ODE in one variable. (Even this last step can be avoided in most
Type I cases.)
The space of realizations is discussed in each case, along with some of their
remarkable geometric properties. Several explicit examples are constructed
(mostly already in the literature) and used to illustrate various features of
the problem.Comment: 43 pages, latex2e with amsart, v2: typos corrected and some minor
improvements in arguments, minor remarks added. v3: important revision,
giving credit for earlier work by others of which the author had been
ignorant, minor typo correction
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