arXiv research

New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.

We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups witho…

Every holomorphic effective parabolic or reductive geometry on a domain over a Stein manifold extends uniquely to the envelope of holomorphy of the domain. This result completes the open problems of my earlier paper on extension of holomorphic geometric structures on complex manifolds. We use this result to classify th…

Paper solves a complex equation for smooth domains.

The study proves a rigidity theorem for convex domains in hyperbolic spaces.

We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed…

We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.

We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.

Deep neural nets solve high-dim PDEs with boundary conditions.

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

Self-affine arcs without inner weak separation are parabolic segments.

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained this way. Further, we investigate …

Study reveals hidden accidental parabolics in complex hyperbolic geometry.

The paper defines capacities for minimal graphs over manifolds and proves the half-space property.

New boundary condition for weak inverse mean curvature flow in bounded domains.

In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the wor…

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, w…

We prove the Bers' density conjecture for singly degenerate Kleinian surfaces groups without parabolics.

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. This extends earlier work don…

We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to $2π/3$.) For the corresponding two-dimensional parabolic free boundary problem we pr…

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M…

We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $μ$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan me…

Smooth solutions up to evolving free boundaries for degenerate equations.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

New domains of discontinuity found for Anosov representations.

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …

This paper concerns the numerical solution of the finite-horizon Optimal Investment problem with transaction costs under Potential Utility. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints. In Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obst…

New BGG sequences on manifolds help solve elasticity and relativity problems.

Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.

Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular (…

Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends…

For a torsion free Kleinian group $Γ$ without parabolics, we consider the decomposition of the limit set $L(Γ)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(Γ)$ when $L(Γ)=S^2_\infty$.

In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the …

A major challenge in current optimization research for deep learning is to automatically find optimal step sizes for each update step. The optimal step size is closely related to the shape of the loss in the update step direction. However, this shape has not yet been examined in detail. This work shows empirically that…

Estimates domain truncation error for option pricing PDEs.

We study actions of discrete subgroups $Γ$ of semi-simple Lie groups $G$ on associated oriented flag manifolds. These are quotients $G/P$, where the subgroup $P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups $Γ\subset G$, we identify domains in oriented flag manifolds by removing a …

Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a st…

The paper extends Huber's theorem to higher dimensions using n-Laplace equations.

Defines connections on parabolic vector bundles for Lie algebroids.

We prove that a finite type curve is an $ξ$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $ξ$-asymptotic line. These results obtained here are generalizations, for plane fields, of the results of V. Arnold [4].

We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear t…