552 research outputs found

    Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

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    We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector

    Histopathological and immunohistochemical evaluation of cellular response to a woven and electrospun polydioxanone (PDO) and polycaprolactone (PCL) patch for tendon repair

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    We investigated endogenous tissue response to a woven and electrospun polydioxanone (PDO) and polycaprolactone (PCL) patch intended for tendon repair. A sheep tendon injury model characterised by a natural history of consistent failure of healing was chosen to assess the biological potential of woven and aligned electrospun fibres to induce a reparative response. Patches were implanted into 8 female adult English Mule sheep. Significant infiltration of tendon fibroblasts was observed within the electrospun component of the patch but not within the woven component. The cellular infiltrate into the electrospun fibres was accompanied by an extensive network of new blood vessel formation. Tendon fibroblasts were the most abundant scaffold-populating cell type. CD45+, CD4+ and CD14+ cells were also present, with few foreign body giant cells. There were no local or systemic signs of excessive inflammation with normal hematology and serology for inflammatory markers three months after scaffold implantation. In conclusion, we demonstrate that an endogenous healing response can be safely induced in tendon by means of biophysical cues using a woven and electrospun patch

    Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids

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    Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.Comment: 36 pages. In this new version the analytic representation hypothesis is proved. Please address all correspondence to D. Peralta-Sala

    Asymptotically cylindrical 7-manifolds of holonomy G_2 with applications to compact irreducible G_2-manifolds

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    We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of the compact G_2-manifolds constructed by Joyce by desingularisation of a flat orbifold T^7/\Gamma can be deformed to one of the compact G_2-manifolds obtainable as a generalized connected sum of two exponentially asymptotically cylindrical SU(3)-manifolds via the method given by the first author (math.DG/0012189).Comment: 36 pages; v2: corrected trivial typos; v3: some arguments corrected and improved; v4: a number of improvements on presentation, paritularly in sections 4 and 6, including an added picture

    Convergence of Ginzburg-Landau functionals in 3-d superconductivity

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    In this paper we consider the asymptotic behavior of the Ginzburg- Landau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via {\Gamma}-convergence, a reduced model for the vortex density, and we deduce a curvature equation for the vortex lines. In a companion paper, we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H_{c1}, and the critical angular velocity of rotating Bose-Einstein condensates.Comment: 45 page

    Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

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    The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

    The Cauchy problems for Einstein metrics and parallel spinors

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    We show that in the analytic category, given a Riemannian metric gg on a hypersurface M⊂ZM\subset \Z and a symmetric tensor WW on MM, the metric gg can be locally extended to a Riemannian Einstein metric on ZZ with second fundamental form WW, provided that gg and WW satisfy the constraints on MM imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an affirmative answer to a question raised in B\"ar, Gauduchon, Moroianu (2005). We also answer negatively the corresponding questions in the smooth category.Comment: 28 pages; final versio
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