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Rigidity theorems for the area widths of Riemannian manifolds
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Rigidity theorems for the area widths of Riemannian manifolds
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The volume spectrum of a compact Riemannian manifold is a sequence of critical values for the area functional, defined in analogy with the Laplace spectrum by Gromov. In this paper we prove that the canonical metric on the two-dimensional projective plane is determined modulo isometries by its volume spectrum. We also prove that the surface Zoll metrics on the three-dimensional sphere are characterized by the equality of the spherical area widths. These widths generalize to the surface case the Lusternik-Schnirelmann lengths of closed geodesics. We prove a new sharp area systolic inequality for metrics on the three-dimensional projective space.
Forward citations
Cited by 3 Pith papers
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Existence of two embedded minimal spheres in $S^3$ with an arbitrary metric
Every Riemannian manifold diffeomorphic to S^{3} contains at least two distinct embedded minimal 2-spheres.
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Equivariant constructions of spheres with Zoll families of minimal spheres
Constructs equivariant one-parameter deformations of S^n (n≥3) admitting Zoll families of minimal spheres, plus first non-linear Zoll metrics on RP^n, via equivariant Nash-Moser-Hamilton IFT.
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Closed minimal surfaces of index one in Riemannian manifolds
Existence of index-one minimal hypersurfaces with unbounded volume in enlargeable manifolds (dims 3-7) plus 3D scalar curvature rigidity under area-nonincreasing maps.
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