6 research outputs found
Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds
We study curvature functionals for immersed 2-spheres in non-compact,
three-dimensional Riemannian manifold without boundary. First, under
the assumption that is the euclidean 3-space endowed with a
semi-perturbed metric with perturbation small in norm and of compact
support, we prove that if there is some point with scalar
curvature then there exists a smooth embedding minimizing the Willmore functional , where
is the mean curvature. Second, assuming that 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
with scalar curvature then there exists a
smooth immersion minimizing the functional , where is the second fundamental form. Finally, adding the
bound to the last assumptions, we obtain a smooth minimizer for the functional . 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
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
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
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