arXiv research

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…

We prove that the second Hochschild cohomology group of the moduli stack of stable $n$-pointed genus $g$ curves vanishes for all but finitely many $(g,n)$.

Study on stable translation lengths of surface homeomorphisms and their approximations.

The paper examines the behavior of Weierstrass measures on stable curves as they approach a nodal stable curve.

We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, bot…

Elliptic curve governs Hopf linking in symmetric tensegrity.

Study shows non-polyhedral structure in moduli spaces for n≥8.

Approximate 3D elastic curves with exact constraints

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…

Study of parabolic Higgs bundles on curves with special fixed points.

We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semi-stable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a c…

The paper proves stability of pulled back parabolic bundles on curves.

Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…

Study variational properties of curves in half-plane with area constraints.

Study the Gromov boundary of fine curve graph for surface homeomorphisms.

Unified approach classifies stable and minimal elastic curves.

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebr…

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…

New proof for stable reduction theorem using Kähler-Einstein metrics.

We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second o…

Stable solutions found for a specific physics model.

Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smootha…

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…

Same cohomology for curves with levels, proving stable range.

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of…

New homotopy theory reveals the structure of stable curves.

Ancient curve shortening flow in a disc with mixed boundary conditions is solved.

We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then the Whitham flow has a singularity. We prove that the stable and unstable manif…

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus …

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…

In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact pseudogroup on the tower. Here, we construct the analogous tower for curves in $n$…

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stie…

Study finds minimal length networks connecting three points in Heisenberg group.

In this article we investigate a family of nonlinear evolutions of polygons in the plane called the $β$-polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable …

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective…

New approach to proving Chen-Donaldson-Sun theorem with examples.

The study classifies points on ruled surfaces in 4-space based on geometric properties.

Stable cylinders found in hyperbolic groups and curve graphs.

Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of stable genus $g$ curves $\bar{\mathcal M}_{g}$. We study the geometry of $Y$, and pr…

New proof shows rationality of scl for non-filling curves.

New findings on stable minimal hypersurfaces in curved 4-manifolds.

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the met…

We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X…

Study real line subbundles on curves, extending classical work.

We prove that the discriminant of a nonsingular space curve of genus $g\geq 2$ is stable with respect to the standard action of the special linear group.

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the co…

An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. F…