1,280 research outputs found

    Generalized Lawson tori and Klein bottles

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    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 33-Sphere and the 44-Sphere

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    We extend the classification of Robert Bryant of Willmore spheres in S3S^3 to variational branched Willmore spheres S3S^3 and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in R3\mathbb{R}^3 and vanishing flux. We also obtain a classification of variational branched Willmore spheres in S4S^4, generalising a theorem of Seb\'{a}stian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved C1,1C^{1,1} 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 33 and 44, such as the sphere eversion, is an integer multiple of 4Ï€4\pi.Comment: 74 pages, 1 figur

    A new construction of homogeneous quaternionic manifolds and related geometric structures

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    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 33-sphere

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    In this article the author investigates flow lines of the classical Willmore flow, which start moving in a parametrization of a Hopf-torus in S3\mathbb{S}^3. 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 CmC^{m}-norm. Moreover, if in addition the Willmore-energy of the initial immersion F0F_0 is required to be smaller than the threshold 8 π338 \, \sqrt{\frac{\pi^3}{3}}, then the unique flow line of the Willmore flow, starting to move in F0F_0, converges fully to a conformal image of the standard Clifford-torus in every CmC^{m}-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration π:S3→S2\pi:\mathbb{S}^3 \to \mathbb{S}^2 w.r.t. the effect of the L2L^2-gradient of the Willmore energy applied to smooth Hopf-tori in S3\mathbb{S}^3 and to smooth closed regular curves in S2\mathbb{S}^2, 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 C\mathbb{C}

    Orientation theory in arithmetic geometry

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    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 â„“\ell-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

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    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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