arXiv research

In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex $T$, the $\mathcal P_rΛ^k(T)$ spaces and the $\mathcal P_r^-Λ^k(T)$ spaces, where $k$ is the degree of the form and $r$ is the degree of its coe…

Symplectic coordinates found on projective structures on orbifolds.

The dynamics of holomorphic 1-forms are studied, showing ergodic foliations and connected spaces.

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite …

In this article, we construct a new para-Kähler structure $({\mathcal G},{\mathcal J},Ω)$ in the space of oriented geodesics ${\mathbb L}(M)$ in a non-flat, real space form $M$. We first show that the para-Kähler metric ${\mathcal G}$ is scalar flat and when $M$ is a 3-dimensional real space form, ${\mathcal G}$ is loc…

The paper defines and analyzes $f$-biharmonic $θ_{α}$-slant curves in $\mathcal{S}$-space forms.

Formula decomposes multiplicative forms on Poisson groupoids into two parts.

Decomposes harmonic forms on almost Kähler manifolds, revealing non-trivial structure.

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $\mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.

It is proved that the isometry classes of pointed connected complete Riemannian $n$-manifolds form a Polish space, $\mathcal{M}_*^\infty(n)$, with the topology described by the $C^\infty$ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifo…

We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\mathbb{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $C$…

This paper explains a 2-form on moduli spaces using quasi-Hamiltonian and Dirac geometry.

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic equation to show that this metric is strictly positive on $\mathcal X$, unless the fam…

The paper proves isomorphisms between two complexes related to singular foliations.

Let $(M,\mathcal{F})$ be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms $Ω^*_b(M,\mathcal{F})$ of the foliation and the "De Rham complex" of the space of leaves $M/\mathcal{F}$ when considered as a "diffeological" quotient. Consequently, the two corresponding …

Outer billiards defined on geodesics surfaces in 3D space forms.

The paper studies webs formed by rational curves on moduli spaces and their abelian relations.

Suppose that we have a compact Kähler manifold $X$ with a very ample line bundle $\mathcal{L}$. We prove that any positive definite hermitian form on the space $H^0 (X,\mathcal{L})$ of holomorphic sections can be written as an $L^2$-inner product with respect to an appropriate hermitian metric on $\mathcal{L}$. We appl…

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $κ=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $Ω^d\mathcal{M}_{g,n}(κ)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $κ$. We…

Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.

Extends Ooguri-Vafa symplectic form to framed Higgs bundles.

The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.

We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analog…

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kähler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangian…

The paper studies Clifford-Bianchi groups acting on hyperbolic spaces and their properties.

The study defines and constructs hypersurfaces in a product of two space forms.

On a closed connected oriented manifold $M$ we study the space $\mathcal{M}_\|(M)$ of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space $\mathcal{M}_\|(M)$ is a smooth …

The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.

The paper proves a pseudo-Kähler structure on a torus's projective space.

The paper explores Hodge decomposition and Hard Lefschetz Condition on almost Kähler manifolds.

New theorem generalizes contact manifolds with symplectic properties.

We define local Hardy spaces of differential forms $h^p_{\mathcal D}(\wedge T^*M)$ for all $p\in[1,\infty]$ that are adapted to a class of first order differential operators $\mathcal D$ on a complete Riemannian manifold $M$ with at most exponential volume growth. In particular, if $D$ is the Hodge--Dirac operator on $…

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping t…

We introduce and analyze a form of variance-reduced $Q$-learning. For $γ$-discounted MDPs with finite state space $\mathcal{X}$ and action space $\mathcal{U}$, we prove that it yields an $ε$-accurate estimate of the optimal $Q$-function in the $\ell_\infty$-norm using $\mathcal{O} \left(\left(\frac{D}{ ε^2 (1-γ)^3} \ri…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in par…

Holomorphic symplectic structure on Lagrangian moduli space.

Extended metric defined on Siegel-Jacobi space using invariant forms.

Calculates volumes of linear subvarieties in moduli spaces of Abelian differentials.

In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in $7$-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.

Study classifies hypersurfaces in 4D Lorentz-Minkowski space using specific operators.

We find the homogenous Kähler isomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the Kähler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$…

Defines a metric and form for a bundle moduli space, leading to a zero-curvature formulation.

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

In the preceding note math.DG/0610917 the $Λ_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In par…

Geometrically classifies total stability spaces for Dynkin diagrams.

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributi…