Yevgeny Liokumovich

Surfaces of small diameter with large width

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Stephane Sabourau.

Spheres of small diameter with long sweep-outs

We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.

Slicing a 2-sphere

We show that for every complete Riemannian surface M diffeomorphic to a sphere with k ≥ 0 holes, there exists a Morse function f : M → ℝ, which is constant on each connected component of the boundary of M and has fibers of length no more than . We also show that on every 2-sphere there exists a simple closed curve of length subdividing the sphere into two discs of area .

Determinantal variety and normal embedding

The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exploiting the conical structure of the stratification of the space of n by n matrices by rank.

Splitting a contraction of a simple curve traversed m times

Suppose that

Sweeping Out 3-Manifold of Positive Ricci Curvature by Short 1-Cycles Via Estimates of Min–Max Surfaces

M

Weyl law for the volume spectrum

is a

Contracting the boundary of a Riemannian 2-disc

2

Existence of minimal hypersurfaces in complete manifolds of finite volume

-dimensional oriented Riemannian manifold, and let

Families of short cycles on Riemannian surfaces

\gamma