arXiv research

Study on Ricci flow on modified Riemann extensions, finding conditions for their preservation.

The paper examines Yamabe flow on modified Riemann extensions and curvature tensors.

Quantizes vortex moduli space using modified Quillen metric.

Generalizations of the Weierstrass formulae to generic surface immersed into $R^4$, $S^4$ and into multidimensional Riemann spaces are proposed. Integrable deformations of surfaces in these spaces via the modified Veselov-Novikov equation are discussed.

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper f…

This note is to concern a generalization to the case of twisted coefficients of the classical theory of Abelian differentials on a compact Riemann surface. We apply the Dirichlet's principle to a modified energy functional to show the existence of differentials with twisted coefficients of the second and third kinds un…

Constructs Bach flat manifolds using modified Riemannian extension.

The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.

New coordinates and Ricci potential formula for moduli space of vector bundles.

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal …

Study describes Yang-Mills flow asymptotics on Riemann surfaces.

The paper explores new structures on cotangent bundles induced by natural Riemann extensions.

We find contact integrable extensions and coverings for the r-th double modified dispersionless Kadomtsev--Petviashvili equation.

Krein's formula for conic Laplacians on compact Riemann surfaces

The subject of this paper is Beurling's celebrated extension of the Riemann mapping theorem \cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurling's geometric and approximative approach. We provi…

Super Riemann surfaces extend Riemann surfaces with an additional field, the gravitino.

Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …

Riemann extension for the anti Mach metric is derived, the solution of geodesic equations for the extended space are given, some properties for the extended space was studied and compared with the basic space and the constructions of a translation surface for the anti Mach metric in four dimension is established.

Extends Floquet-Bloch theory to nilpotent groups for geometric applications.

The paper studies metric connections with torsion on cotangent bundles with modified Riemannian extensions.

Study on null-projectability of Levi-Civita connections in neutral metrics.

Study spherical conic metrics on Riemann surfaces with isolated singularities.

New topological Riemann-Roch theorem for circle fibrations.

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,φ_0)$ converges to a critical point $(A_\infty, φ_\infty)$ of this functional. Using a modified Chern-Wei…

New convex programs solve minimal-area problems on Riemann surfaces.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

Some examples of three-dimensional metrics of constant curvature defined by solutions of nonlinear integrable differential equations and their generalizations are constructed. The properties of Riemann extensions of the metrics of constant curvature are studied. The connection with the theory of normal Riemann spaces a…

In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory. We generalize their study to all closed, compact, connected, possibly nonorientable surfaces. We introduce the notion of "super central extension" of the fund…

Constructs hyper-Kähler models using Riemann-Hilbert problems.

This note provides a new proof of the real analyticity of the Liouville map.

Synthetic Differential Geometry modifies local space structures with infinitesimal curvature.

Constructs a map from stable extensions to irreducible metrics on Riemann surfaces.

A general scheme for construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to WDVV associativity equations is formulated. The advantage is taken from the Rota-Baxter identity and some relation being counterpart of the modified Yang-Baxter identity from the classical …

The paper proves formulas for determinant determinants of Laplacians on Riemann surfaces with conical singularities.

Study finds determinant of Laplacian on a special surface with a conical point.

Constructs a connection for Hodge theoretic projective structures on Riemann surfaces.

Consider a smooth manifold with a smooth metric which changes bilinear type from Riemann to Lorentz on a hypersurface $Σ$ with radical tangent to $Σ$. Two natural bilinear symmetric forms appear there, and we use it to analyze the geometry of $Σ$. We show the way in which these forms control the smooth extensibility ov…

Paper introduces deterministic EM approximations for non-convex likelihood functions.

The Kazdan-Warner problem is solved for Riemann surfaces with smooth boundaries.

New formalization of curved spaces using pointwise affine spaces.

Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla$ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $$ on $T^{\ast}M$ defined by means of a symmetric $$-tensor field $c$ on $M.$…

The paper develops a Galois theory for cluster algebras and Riemann surfaces.

We apply the technique of integrable extensions to the symmetry pseudo-group of the r-th mdKP equation. This gives another look on deriving known coverings and allows us to find new coverings for this equation.

Study the determinant of Laplacians on Riemann surfaces with conical metrics.

In this paper, we solve the optimal constant problem in the setting of Ohsawa's generalized $L^{2}$ extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces…

Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated.

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…