Related papers: Manickam-Mikl\'os-Singhi Conjectures on Partial Ge…
We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic…
The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…
The main result of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (non necessarily pseudo-Riemannian) in terms of the existence of certain connection. The result generalizes…
The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction…
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the…
In this paper, we prove the coarse geometric $\ell^p$-Novikov Conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.
The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not…
In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…
We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq…
Some geometric structures with associated Riemannian metrics have been considered in the book.
This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a…
The Ramanujan conjecture for modular forms of holomorphic type was proved by Deligne almost half a century ago: the proof, based on his earlier proof of Weil's conjectures, was an achievement of algebraic geometry. We give here a short…
We discuss the connection between the smooth and metric structure on quotient spaces, prove smoothness of isometries in special cases and discuss an application to a conjecture of Molino.
We determine when a matrix is similar to a partial isometry, refining a result of Halmos--McLaughlin.
The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Mikl\"os and Singhi. We study the fundamentals properties of such lattices and of a…
A surface M is called p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the twodimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
We prove that the partial $C^0$-estimate holds for metrics along Aubin's continuity method for finding K\"ahler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of…
The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is…