arXiv research

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

In this paper we classify maps from a torus phase space $X$ to $\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain symmetry class. Furthe…

We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point with…

This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…

In this paper, we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds. In the case of a family of smooth Calabi-Yau hypersurfaces in projective space degenerating into the transversal union of two smooth Fano hypersurfaces in a generic …

In this paper we show that two Lagrangian graphs over the torus in $\mathbb{C}^n$ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the $L^2$ metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in $\mathbb{C}^n$ ca…

Let $S$ be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of $S$ lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when $S$ becomes Euclidean, i.e. very small.

In 2009 Gaiotto, Moore and Neitzke presented a new construction of hyperkähler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration $\mathcal{M}$ over a base space $\mathcal{B}$, except for a divisor $D$ in $\mathcal{B}$, in which the torus fiber degenerates into a nodal t…

Two non-Morse-Bott Chern-Simons functions on homology 3-spheres.

For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in defo…

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $8π$. In particular, every constrained Willmore torus with Willmore energy below $8π$ and non-rectangular conformal class is non-degenerated.

We describe a simple way of constructing torus fibrations $T^3\to X\to S^3$ which degenerate canonically over a knot or link in $S^3$. We show that the topological invariants of $X$ can be computed algebraically from the monodromy representation of the fibration. We use this to obtain some new $T^3$-fibrations $S^3\tim…

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures $(M,I,J',K')$ containing $I$…

We present a new infinite class of near-horizon geometries of degenerate horizons, satisfying Einstein's equations for all odd dimensions greater than five. The symmetry and topology of these solutions is compatible with those of black holes. The simplest examples give horizons of spatial topology S^3xS^2 or the non-tr…

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ where a short closed geodesic is pinched. If the geodesic separates the surface into two parts, then the Jacobian torus of $S$ develops into a torus that splits. If the geodesic is nonseparating then the Jacobian torus of $…

Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that…

We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, …

The paper explores hidden torus symmetries in integrable systems and their stability.

Counterexample disproves conjecture on flat metrics and fiber bundles.

The study simplifies complex functions on surfaces using a special transformation.

We consider Lorentzian manifolds with parallel light-like vector field V. Being parallel and light-like, the orthogonal complement of V induces a codimension one foliation. Assuming compactness of the leaves and non-negative Ricci curvature on the leaves it is known that the first Betti number is bounded by the dimensi…

Let $X_n$ be a cycle of $n$ projective lines, and $\bT_n$ a symplectic torus with $n$ punctures. Using the theory of spherical twists introduced by Seidel and Thomas (2001), I will define an action of the pure mapping class group of $\bT_n$ on $D^b(Coh(X_n))$. The motivation comes from homological mirror symmetry for d…

The study classifies compact Cauchy horizons in vacuum spacetimes.

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…

Computes knot filtered ECH for torus knots on tight 3-sphere.

Let T be a complex torus, and X the surface CP^1 x T. If T is embedded in CP^{n-1} then X may be embedded in CP^{2n-1}. Let X_Gal be its Galois cover with respect to a generic projection to CP^2. In this paper we compute the fundamental group of X_Gal, using the degeneration and regeneration techniques, the Moishezon-T…

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to…

This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. …

Let $φ: S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $φ$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $φ$ has no critical points at all). We introduce …

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributiona…

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical vari…

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. …

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We…

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…

Let $X$ be a compact toric Kähler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula…

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.

We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle $Q$, the degenerate homology of $Q$ is completely determined by the quandle homology of $Q$. For this case (and generally for two term homology of …

We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singula…

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the foca…

Study degenerate Bianchi transformations for pseudo-spherical submanifolds in 5D space.

New stabilization found in planar elasticae with degenerate diffusion.

Uniform estimates for Calabi-Yau degenerations proved.

Sharp diameter bounds for Calabi-Yau degenerations proved.

Paper solves degenerated circle pattern metric problem in spherical geometry.