Related papers: (Weak) $m$-extremals and $m$-geodesics

In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball,…

The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved…

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with…

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved…

There were obtained some properties of weakly m-convex sets. Various kinds of the problem of shadow were investigated. There was obtained the lower estimation for the number of balls that are necessary to create a shadow at the point of the…

In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…

We study quasiconvex quadratic forms on $n \times m$ matrices which correspond to nonnegative biquadratic forms in $(n,m)$ variables. We disprove a conjecture stated by Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017) as well as…

For a compact Riemannian $3$-manifold $(M^{3}, g)$ with mean convex boundary which is diffeomorphic to a weakly convex compact domain in $\mathbb{R}^{3}$, we prove that if scalar curvature is nonnegative and the scaled mean curvature…

In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and…

We show that, in the Teichm\"uller metric, "thin-framed triangles are thin"---that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on…

We prove that all normalized symplectic capacities coincide on smooth domains in $\mathbb C^n$ which are $C^2$-close to the Euclidean ball, whereas this fails for some smooth domains which are just $C^1$-close to the ball. We also prove…

The present work considers the properties of generally convex sets in the $n$-dimensional real Euclidean space $\mathbb{R}^n$, $n>1$, known as weakly $m$-convex, $m=1,2,\ldots,n-1$. An open set of $\mathbb{R}^n$ is called weakly $m$-convex…

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichm\"uller spaces, and hyperbolic 3-manifolds. In the same paper, they…

We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $\mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a…

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in $W^{m,\frac{n}{m}}_{\mathcal{N}}(\Omega)$,…

We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $\alpha$, the length of…

We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be…

By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in…