arXiv research

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

The paper studies connections in superintegrable systems, revealing geometric insights.

Proves a conjecture for Calabi-Yau manifolds.

Minimal surfaces in hyperbolic 4-space degenerate to core of product trees.

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…

Study Riemannian geometry of maximal surface group representations in pseudo-hyperbolic space.

In this note we study the localization of Futaki-Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implement the localization procedure for a holomorphic vector field on $CP^n$ with a maximally degenerate zero, giving an essentially unique formula for the Futaki-…

Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichmüller space of $S$, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponen…

For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between t…

New method resolves nonidentifiability in mixture models.

Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.

A class of codimension one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally non-degenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson type submanifold. In particular we deduce that their leaf spaces are homeomor…

Superintegrable systems on curved manifolds found to have Hessian structures.

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's cl…

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…

Let $S$ be a closed oriented surface of genus at least $2$. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over the Teichmüller space of $S$, we study the behaviour of these geometric structures along pinching…

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compacti…

Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.

The paper explores optimization of higher Steklov eigenvalues in various dimensions.

For an almost product structure $J$ on a manifold $M$ of dimension $6$ with non-degenerate Nijenhuis tensor $N_J$, we show that the automorphism group $G=Aut(M,J)$ has dimension at most 14. In the case of equality $G$ is the exceptional Lie group $G_2^*$. The next possible symmetry dimension is proved to be equal to 10…

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 construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work…

New geometric transitions studied via real algebra degenerations.

Compactifies a component by studying metric degeneration.

Study fibrations of projective spaces for maximal representations.

Study finite-energy metrics over complex manifold degenerations.

The paper classifies second-order superintegrable systems with torsion and semi-degeneracy.

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group $G(M)$ of automorphisms of $M$. We characterize also the standa…

We consider the value function originating from an expected utility maximization problem with finite fuel constraint and show its close relation to a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. On one hand, we give a so-called verification argument based on the dynamic progr…

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\d…

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and a…

Maximizes coding rate difference for robust, discriminative features.

Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension $(2n+1)$ have maximal symmetry dimension $n^2+4n+3$. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is $n^2+4$ for Levi-indefinite structures and $n^2+3$ for Levi-definite structures when $n>1…

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

This paper improves tail dependence analysis by introducing a path-based approach.

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric …

Maximizing energy on flexible curves yields regular or convex polygons.

Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G…

New superintegrable systems derived from Frobenius structures.

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the su…

This work takes up the challenges of utility maximization problem when the market is indivisible and the transaction costs are included. First there is a so-called solvency region given by the minimum margin requirement in the problem formulation. Then the associated utility maximization is formulated as an optimal swi…

This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not nec…

Construct special Lagrangian fibrations on abelian varieties using retraction techniques.