arXiv research

New method calculates cut locus on surfaces without boundary.

Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

Totally geodesic subvarieties in moduli spaces are studied.

Machine learning identifies boundaries of real solutions in polynomial systems.

Capacity control, the bias/variance dilemma, and learning unknown functions from data, are all concerned with identifying effective and consistent fits of unknown geometric loci to random data points. A geometric locus is a curve or surface formed by points, all of which possess some uniform property. A geometric locus…

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the com…

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The …

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular loc…

In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this pap…

We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually ordinary link concordance invariants. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.

Anti-de Sitter spacetimes with convex boundaries are uniquely determined by their boundary metrics.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus…

Study continuity and Hölder estimates for solutions on Stein spaces.

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…

The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove…

Let $\mathcal M_g$ denote the moduli space of compact Riemann surfaces of genus $g$ and let $\mathcal A_g$ be the space of principally polarized abelian varieties of (complex) dimension $g$. Let $J:\mathcal M_g\longrightarrow \mathcal A_g$ be the map which associates to a Riemann surface its Jacobian. The map $J$ is in…

In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product $A_1 \times A_2$, where one of the factors admits real multiplication by a real quadratic order $\mathcal{O}_D$ of discriminant $D$. We show that the moduli space $X_D^{(3)}$ of these varieties essentially is the disjoint unio…

We show that a four-manifold admits a boundary Lefschetz fibration over the disc if and only if it is diffeomorphic to $S^1 \times S^3\# n \overline{\mathbb{C} P^2}$, $\# m\mathbb{C} P^2 \#n\overline{\mathbb{C} P^2}$ or $\# m (S^2 \times S^2)$. Given the relation between boundary Lefschetz fibrations and stable general…

Study foliations at infinity and constant mean curvature surfaces in quasi-Fuchsian manifolds.

Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.

In this paper, we study the boundary behavior of the negatively curved Kähler-Einstein metric attached to a log canonical pair $(X,D)$ such that $K_X+D$ is ample. In the case where $X$ is smooth and $D$ has simple normal crossings support (but possibly negative coefficients), we provide a very precise estimate on the p…

The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions, each member of which is called an {\it almost distance function}. Typical examples of an almost distance function are the distance function…

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

Study non-existence of complex ball quotients in Torelli locus.

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the …

We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split loc…

The paper describes the CR umbilical locus of a real ellipsoid in complex space.

New 2-spheres of revolution with simple cut locus structures.

Study on Blaschke locus with covariance metric properties.

The paper studies geometric loci and their invariants in complex dynamics.

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

Study conjugate locus in convex 3-manifolds using Jacobi fields.

Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.

The paper extends spacetime topology results using codimension 2 null cut locus properties.

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichmüller space is represented as a path $γ$, we show that $γ$ joins the …

Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…

We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the distinct, orientable, cut locus structures on graphs with four generating cycles.

We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …

The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point $0$ is equal to the set of coheren…

Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.

Study on cut locus of submanifolds in Finsler geometry.

In the previous paper, the structure of the cut locus was determined for a class of surfaces of revolution homeomorphic to a cylinder. In this paper, we prove the structure theorem of the cut locus for a wider class of surfaces of revolution homeomorphic to a cylinder.

Study of twistor spaces and minitwistor spaces for ALE gravitational instantons.

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compres…