arXiv research

Proves a higher rank rigidity theorem for convex real projective manifolds.

Proves a theorem for Assouad dimension with applications to distance sets and radial projections.

We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of hyperbolic $n$-space by a geometric argument. Moreover, we obtain a Besicovitch-Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.

New proof of four squares theorem using projective geometry.

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-or…

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

Entropy rigidity proven for 3D and higher convex projective manifolds.

We discuss eight new(?) configuration theorems of classical projective geometry in the spirit of the Pappus and Pascal theorems.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

3D projective structures can be metrized with conformal structures.

The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.

In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed $C^r$ mappings are shown $(r\geq 1)$. Moreover, some applications of the two the…

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

The paper classifies certain singular projective varieties with specific properties.

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…

New extension theorem for projective manifolds.

A new projective structure on Riemann surfaces differs from the uniformization theorem's structure.

These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of t…

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a…

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.

Combination theorems for convex projective geometry subgroups.

Study on volumes of random inscribed polytopes in projective geometries.

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…

The authors give a complete classification of projective threefolds admitting a holomorphic normal projective connection. Moreover, they prove a general structure theorem on complex projective manifolds admitting a holomorphic normal projective connection, saying in particular, that any such manifold is either the proj…

We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces.

Proves projectability of $H$-surfaces in non-perpendicular boundary conditions.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.

This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.

Extends Tian theorem to Vaisman manifolds for approximations.

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.

Weyl-type theorems extended to Galilei and Carroll geometries.

Study smooth embeddings of line configurations in complex projective plane.

The study classifies stable submanifolds in product spaces of projective spaces.

Theory of link projections to 3-manifold spines, proving combinatorial moves for isotopic links.

When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor p…

The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential ope…

The study connects projective codes to the distribution of zeros of odd maps.

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces.

Symplectic embedding extended to stratified spaces.

We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to t…

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.

Solves open problems on curved projective varieties.