Proves inequality for 1-dimensional cycles.
problem Proving the Parametric Coarea Inequality for 1-cycles.
method Analytical proof based on conjecture by Guth and Liokumovich.
result Proved the Parametric Coarea Inequality for 1-cycles.
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16 results for “Liokumovich”
For almost all Riemannian metrics (in the Baire sense) on a closed manifold , , we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in . This gives a quantitative version of the main result of \cite{irie-marques…
The paper shows equidistribution of geodesics and nets on manifolds.
problem Equidistribution of geodesics and nets on manifolds.
method Generic metrics, Weyl Law for volume spectrum, and equidistribution of geodesic nets.
result Equidistribution of geodesic nets on 3-manifolds.
Contracting thin disksmath.DG
We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by .
The p-widths of a surfacemath.DG
New insights into -widths of surfaces, proving optimality and calculating constants.
problem Understanding -widths of surfaces and their relationship to geodesics.
method Analyzing min-max sequences of minimal submanifolds and sweepouts of surfaces.
result Optimal sweepouts of the round two-sphere by great circles.
Generic smooth minimal hypersurfaces exist in 8D manifolds.
problem Existence of smooth minimal hypersurfaces in high-dimensional manifolds.
method Global perturbation argument and a novel geometric invariant.
result Generic metrics on 8D manifolds admit smooth minimal hypersurfaces.
We show that a Weyl law holds for the variational spectrum of the -Laplacian. More precisely, let be the variational spectrum of on a closed Riemannian manifold and let be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorna…
Uryson width and volumemath.DG
We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich-Lishak-Nabutovsky-Rotman. We show also that for any there is a Riemannian metric on a 3-sphere such that and for an…
Weyl law for 1-cyclesmath.DG
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
New bounds on shortest geodesic loops on a sphere.
problem Finding shortest geodesic loops on a sphere.
method Analyzing geodesic loops starting and ending at a fixed point on a sphere.
result At any point on a sphere, there are at least two distinct geodesic loops whose lengths are bounded by 8d and 14d.
The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.
problem Finding the shortest geodesic flower on a specific class of manifolds.
method Analyzing a non-compact Riemannian manifold with locally convex ends and finite volume, proving the existence of a geodesic net with constraints on its length.
result The existence of a non-trivial geodesic flower with a bounded total length on the manifold.
Proves non-existence of certain metrics on specific manifolds.
problem Existence of metrics with nonnegative scalar curvature.
method Connected sum technique and non-compact domination method.
result Connected sum of aspherical manifolds with non-compact manifolds does not admit metrics with nonnegative scalar curvature.
We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We …
Study proves properties of constant mean curvature hypersurfaces in high-dimensional spaces.
problem Properties of constant mean curvature hypersurfaces in high-dimensional spaces.
method Proves properties of constant mean curvature hypersurfaces using min-max procedure and surgery.
result Every tangent cone at each isolated singularity is area-minimising.
Closed geodesics found on specific non-compact manifolds.
problem Existence of closed geodesics on non-compact Riemannian manifolds.
method Conditions on the manifold and its boundary to ensure the existence of closed geodesics.
result Existence of non-trivial closed geodesics on the manifold.
The paper proves existence of solutions to the Allen-Cahn equation on certain Riemannian manifolds.
problem Existence of finite energy solutions to the Allen-Cahn equation on complete Riemannian manifolds of finite volume.
method Proves the existence of solutions using the energy method and properties of the ambient metric.
result For a wide range of ε, there exists a finite energy solution to the Allen-Cahn equation on a complete Riemannian manifold of finite volume.