868 research outputs found
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
Weyl law for the volume spectrum
Given a Riemannian manifold with (possibly empty) boundary, we show that
its volume spectrum satisfies a Weyl law
that was conjectured by Gromov.Comment: Revised version. To appear in Annals of Mathematic
Rigidity of min-max minimal spheres in three-manifolds
In this paper we consider min-max minimal surfaces in three-manifolds and
prove some rigidity results. For instance, we prove that any metric on a
3-sphere which has scalar curvature greater than or equal to 6 and is not round
must have an embedded minimal sphere of area strictly smaller than and
index at most one. If the Ricci curvature is positive we also prove sharp
estimates for the width
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