arXiv research

Formula for sectional curvature on 2D Lorentzian manifolds derived.

Study of singularities in two-dimensional Nijenhuis operators with non-zero trace differential.

We give a survey on the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional and on related problems of spectral theory of the two-dimensional Dirac operator with periodic coefficients.

Formula derived for curvature on smooth manifolds.

Two-dimensional Lagrangian mean curvature equation solved with new inequality.

Develops probabilistic forecasting for Sea Level Anomalies using Conformal Prediction on functional time series.

We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.

For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvat…

The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due t…

Solves an Arnold trivium problem using calculus and topology.

Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.

We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R^3. This hypothesis was formulated recently by I.A.Taimanov (dg-ga/9610013). Hi…

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler--Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calc…

We study orthonormal normal sections of two-dimensional immersions in $\mathbb R^{n+2},$ $n\ge 2$, at which these sections are critical for a functional of total torsion. In particular, we establish upper bounds for the torsion coefficients in the case of non-flat normal bundles. With these notes we continue a foregoin…

Lectures detail field theory dynamics and exact WKB analysis.

2D CNNs approximate Korobov functions with near-optimal rates.

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action f…

We show that geometric integrals of the type $\int_Ωf\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $Ω$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of $Ω$ has sufficiently small dimensio…

It is shown that any two-dimensional spacetimes with compact Cauchy surfaces can be causally isomorphically imbedded into the two-dimensional Einstein's static universe. Also, it is shown that any two-dimensional globally hyperbolic spacetimes are conformally equivalent to a subset of the two-dimensional Einstein's sta…

Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio.

New geometries explain solutions to differential equations.

We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group $G$ associated to outer automorphisms of $G$, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted $G$-bundles, and calculate it ex…

We obtain various estimates of the life-time of two-dimensional minimal tubes in R^3 by potential theory methods.

We consider a two-dimensional optimal dividend problem in the context of two branches of an insurance company with compound Poisson surplus processes dividing claims and premia in some specified proportions. We solve the stochastic control problem of maximizing expected cumulative discounted dividend payments (among al…

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…

For two-dimensional, immersed closed surfaces $f:Σ\to \R^n$, we study the curvature functionals $\mathcal{E}^p(f)$ and $\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asser…

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded curvature. Second, we give examples of complete two-dimensional expanding Ricci solitons …

Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.

Two-dimensional case in the theory of dynamical systems admitting the normal shift differs crucially from multidimensional case. Features of two-dimensional case are gathered and studied in this thesis.

Researchers found abnormal extremals on specific Lie groups.

We give a condition under which the findings of the paper cited above work well and determine the surfaces that were not considered before. In this paper, we show that a parallel mean curvature surface of a general type in a complex two-dimensional complex space form depends on one real-valued harmonic function on the …

All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples which I obtained in "Moebius structures and two dimensional Einstein-Weyl geometry" J. reine angew. Math. 504 (1998).

A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Fin…

We introduce a new local sparse attention layer that preserves two-dimensional geometry and locality. We show that by just replacing the dense attention layer of SAGAN with our construction, we obtain very significant FID, Inception score and pure visual improvements. FID score is improved from $18.65$ to $15.94$ on Im…

We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as …

The excluded area between a pair of two-dimensional hard particles with given relative orientation is the region in which one particle cannot be located due to the presence of the other particle. The magnitude of the excluded area as a function of the relative particle orientation plays a major role in the determinatio…

In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is exactly the cubic form. We also consider parameterizations on two dimensional affi…

Generalizes Toponogov theorem to Alexandrov spaces.

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of varia…

We prove a local graphical theorem for two-dimensional self-shrinkers away from the origin. As applications, we study the asymptotic behavior of noncompact self-shrinkers with finite genus. Also, we show uniform boundedness on the second fundamental form of two-dimensional noncompact self-shrinkers with bounded mean cu…

In this paper, we study a class of two-dimensional Finsler metrics defined by a Riemannian metric $α$ and a 1-form $β$. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that $β$ is always closed for those metrics in higher dim…

New method approximates Gaussian curvature on discrete surfaces.

We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqu…

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend o…

The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauc…

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillar…

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.