381 research outputs found
Minimal surfaces - variational theory and applications
Minimal surfaces are among the most natural objects in Differential Geometry,
and have been studied for the past 250 years ever since the pioneering work of
Lagrange. The subject is characterized by a profound beauty, but perhaps even
more remarkably, minimal surfaces (or minimal submanifolds) have encountered
striking applications in other fields, like three-dimensional topology,
mathematical physics, conformal geometry, among others. Even though it has been
the subject of intense activity, many basic open problems still remain. In this
lecture we will survey recent advances in this area and discuss some future
directions. We will give special emphasis to the variational aspects of the
theory as well as to the applications to other fields.Comment: Proceedings of the ICM, Seoul 201
Applications of almgren-pitts min-max theory
We develop an application of Almgren-Pitts min-max theory to the study of minimal hypersurfaces in dimension 3 ≤ m + 1 ≤ 7 as well as computing the k-width of the round 2-sphere for k = 1,...,8. We show that the space of minimal hypersurfaces is non-compact for an analytic metric of positive curvature and construct a min-max unstable closed geodesic in S^2 with multiplicity 2.Open Acces
On the multiplicity of isometry-invariant geodesics on product manifolds
We prove that on any closed Riemannian manifold , with
\rank\Hom_1(M_1)\neq0 and , every isometry homotopic to the
identity admits infinitely many isometry-invariant geodesics.Comment: 17 page
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