6 research outputs found

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

    Get PDF
    We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold (M,h)(M,h) without boundary. First, under the assumption that (M,h)(M,h) is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in C1C^1 norm and of compact support, we prove that if there is some point xˉ∈M\bar{x} \in M with scalar curvature RM(xˉ)>0R^M(\bar{x})>0 then there exists a smooth embedding f:S2↪Mf:S^2 \hookrightarrow M minimizing the Willmore functional 1/4∫∣H∣21/4\int |H|^2, where HH is the mean curvature. Second, assuming that (M,h)(M,h) is of bounded geometry (i.e. bounded sectional curvature and strictly positive injectivity radius) and asymptotically euclidean or hyperbolic we prove that if there is some point xˉ∈M\bar{x} \in M with scalar curvature RM(xˉ)>6R^M(\bar{x})>6 then there exists a smooth immersion f:S2↪Mf:S^2 \hookrightarrow M minimizing the functional ∫(1/2∣A∣2+1)\int (1/2|A|^2+1), where AA is the second fundamental form. Finally, adding the bound KM≤2K^M \leq 2 to the last assumptions, we obtain a smooth minimizer f:S2↪Mf:S^2 \hookrightarrow M for the functional ∫(1/4∣H∣2+1)\int (1/4|H|^2+1). The assumptions of the last two theorems are satisfied in a large class of 3-manifolds arising as spacelike timeslices solutions of the Einstein vacuum equation in case of null or negative cosmological constant.Comment: 19 Page

    Willmore minimizers with prescribed isoperimetric ratio

    Full text link
    Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio

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

    Full text link
    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

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

    No full text
    Abstract We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M . Under the assumption that the sectional curvature K M is strictly positive, we prove the existence of a smooth immersion f : S 2 → M minimizing the L 2 integral of the second fundamental form. Assuming instead that K M ≤ 2 and that there is some point x ∈ M with scalar curvature R M (x) > 6, we obtain a smooth minimizer f : S 2 → M for the functional 1 4 |H| 2 + 1, where H is the mean curvature. Key Words: L 2 second fundamental form, Willmore functional, direct methods in the calculus of variations, geometric measure theory, elliptic regularity theory
    corecore