4,040 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
Morse index and multiplicity of min-max minimal hypersurfaces
The Min-max Theory for the area functional, started by Almgren in the early
1960s and greatly improved by Pitts in 1981, was left incomplete because it
gave no Morse index estimate for the min-max minimal hypersurface.
We advance the theory further and prove the first general Morse index bounds
for minimal hypersurfaces produced by it. We also settle the multiplicity
problem for the classical case of one-parameter sweepouts.Comment: Cambridge Journal of Mathematics, 4 (4), 463-511, 201
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian
three-manifold admits an infinite number of closed immersed minimal surfaces.
We use min-max theory for the area functional to prove this conjecture in the
positive Ricci curvature setting. More precisely, we show that every compact
Riemannian manifold with positive Ricci curvature and dimension at most seven
contains infinitely many smooth, closed, embedded minimal hypersurfaces.
In the last section we mention some open problems related with the geometry
of these minimal hypersurfaces.Comment: 34 pages, to appear in Inventiones Mathematica
Min-max theory and the energy of links
Freedman, He, and Wang, conjectured in 1994 that the Mobius energy should be
minimized, among the class of all nontrivial links in Euclidean space, by the
stereographic projection of the standard Hopf link. We prove this conjecture
using the min-max theory of minimal surfaces.Comment: 19 pages. Revised version. To appear in J. Amer. Math. So
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