Related papers: Convex Integration and Legendrian Approximation of…

The original Arnold chord conjecture states that every closed Legendrian submanifold of the standard contact sphere $S^{2n-1}$ admits a Reeb chord with distinct endpoints with respect to any contact form. In this paper, we prove this…

This note offers a first bridge from machine learning to modern differential geometry. We show that the logits-to-probabilities step implemented by softmax can be modeled as a geometric interface: two potential-generated, conservative…

We consider several natural sets of curves associated to a given Teichm\"uller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to…

We construct the first smooth embedded compact special Legendrian surfaces in \(\mathbb S^5\) of genus greater than one. More precisely, for every sufficiently large integer \(k\), we construct an embedded special Legendrian surface whose…

In this paper, by refining approximation theorems for holomorphic sections of adjoint line bundles, it is proved that the regular locus of a weakly pseudoconvex complex space admitting a positive line bundle can be holomorphically embedded…

We define a relative version of contact homology for contact manifolds with convex boundary, and prove basic properties of this relative contact homology. Similar considerations also hold for embedded contact homology.

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied and its Euler-Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature…

Suppose that L is a null--homologous Legendrian knot in a contact 3--manifold. We determine the connection between the sutured invariant of the complement of L and the Legendrian invariant defined by Lisca, Ozsvath, Stipsicz and Szabo. In…

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a…

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…

We approximate boundaries of convex polytopes by smooth hypersurfaces $Y=Y_\varepsilon$ with {\it positive mean curvatures} and, by using basic geometric relations between the scalar curvatures of Riemannin manifolds and the mean curvatures…

Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic…

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic…

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding…

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold $(V, \xi)$ has…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

A theorem of Ding and Geiges states that every closed, connected contact $3$-manifold can be obtained from the standard tight contact $3$-sphere by contact $(\pm1)$-surgery along a Legendrian link. The literature also contains some examples…

Reconstructing 3D scenes from multiple views has made impressive strides in recent years, chiefly by correlating isolated feature points, intensity patterns, or curvilinear structures. In the general setting - without controlled…