arXiv research

Study Blaschke's asymptotic lines on surfaces in 3D space.

The paper examines asymptotic lines of plane fields in 3D space.

Master thesis proves Bergman kernel asymptotics for positive line bundles.

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…

Study of torsion forms for positive line bundles.

Study of Moncrief lines' behavior in curved space-times.

Study stability thresholds of big line bundles, proving bounds and generalizing results.

We prove that a finite type curve is an $ξ$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $ξ$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].

In this paper are given examples of tori T^2 embedded in S^3 with all their asymptotic lines dense.

Geometric quantization extended to big line bundles.

Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.

Geodesic lines with specific boundaries found on a special type of manifold.

Researchers found infinitely many non-collapsed steady Ricci solitons on complex line bundles.

Formula derived for a magnetic line invariant.

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replac…

Study on the asymptotic geometry of Higgs bundles over projective line.

The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…

The study defines invariants for time-like surfaces with real asymptotic lines.

In this paper we study some properties of surfaces immersed in $\mathbf R ^4$ whose asymptotic lines are orthogonal. We also analyze necessary and sufficient conditions for the hypersphericity of surfaces in $\mathbf R ^4$.

Extended Einstein manifolds reveal new symmetries.

The paper establishes conditions for optimal sampling configurations on complex manifolds.

Study complex lines in symplectic geometry, generalizing previous results.

Study optimal holomorphic extensions for jets along submanifolds as tensor powers increase.

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…

The paper proves heat kernel asymptotics for high power line bundles on complex manifolds.

We find the entropy's infinite-size behavior in complex manifold sections.

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal…

The paper analyzes systemic risk in an insurance model with multiple business lines and heterogeneous claims.

Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.

As a consequence of a result of Cardoso and Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line. The case of non-trapping metrics with constant curvature near infinity is also considered: there exists a strip with at …

Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lin…

New method for spectral and Bergman kernels under local spectral gap condition.

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic exp…

We describe the structure of the asymptotic lines near an inflection point of a Lagrangean surface, proving that in the generic situation it corresponds to two of the three possible cases when the discriminant curve has a cusp singularity. Besides being stable in general, inflection points are proved to exist on a comp…

We study the asymptotic of the Bergman kernel of the spin$^c$ Dirac operator on high tensor powers of a line bundle.

We provide a proof and analyze the asymptotic behavior of a formula for the linking number of line segments.

We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $α$-curve shortening flow for exponents $α>\frac12$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $α$-curve shortening flow to the …

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

The Teichmüller space $\mathcal{T}(Σ)$ of a surface $Σ$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(Σ)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.

Surveying random sections on Kähler manifolds, leading to metrics.

The paper studies volumes of direct images for high tensor powers of ample bundles.

We study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the o…

The paper examines the geometry of a curve's centre symmetry set.

We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $\R^3$. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander…