arXiv research

The paper examines stability of harmonic and symphonic maps with forms and potentials.

Proves existence of global positive LCK potential on certain manifolds.

Harmonic spinors linked to twistor spinors via potential forms and conformal Killing-Yano forms.

We discuss critical elliptic systems in potential form. We prove existence, multiplicity, and compactness of solutions.

New theorem on Lee classes for LCK manifolds with potential.

It is shown that an HKT-space with closed parallel potential 1-form has $D(2,1;-1)$-symmetry. Every locally conformally hyperkähler manifold generates this type of geometry. The HKT-spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic conditio…

Defines new extremal potentials and measures for Kähler forms.

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence …

Develops a framework for potentials on Lauritzen manifolds using bi-forms.

In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a …

The paper studies special solitons on Riemannian manifolds with specific vector fields.

Extracts interpretable potential energy from Hamiltonian systems.

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…

Study on real hypersurfaces with special solitons in complex space forms.

Study the full bosonic string action in Minkowski space to Riemannian manifolds.

Upper and lower bounds for magnetic Laplacian eigenvalues on manifolds.

New potential functions reveal Frobenius-like structure in weighted hyperplane arrangements.

Researchers find a way to estimate potential functions for quaternionic metrics.

The paper describes flat Hessian metrics on surfaces and their potentials.

New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.

The study introduces a new function to analyze special holonomy manifolds.

Bi-forms extend contrast functions to handle torsion in information geometry.

Topological field theories of 2- and 3-forms in 6D, showing no propagating degrees of freedom.

Study on Ricci-like solitons on specific geometric manifolds.

Study on almost Riemann solitons with gradient or torse-forming vector fields.

Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we fi…

Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-Δ$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\nabla^*\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bou…

Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.

New potentials found for sheaves on Calabi-Yau 4-folds.

We introduce a class of potential submanifolds in pseudo-Euclidean spaces (each N-dimensional potential submanifold is a special flat torsionless submanifold in a 2N-dimensional pseudo-Euclidean space) and prove that each N-dimensional Frobenius manifold can be locally represented as an N-dimensional potential submanif…

We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…

Study on a new type of solitons on specific geometric manifolds.

We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an aux…

Study existence of harmonic 1-forms on Calabi-Yau manifolds.

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

Study of Yamabe solitons on specific geometric manifolds.

The paper bounds eigenvalues of magnetic Schroedinger operators on compact manifolds.

Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there …

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is…

The caloron correspondence is a tool that gives an equivalence between principal $G$-bundles based over the manifold $M \times S^1$ and principal $LG$-bundles on $M$, where $LG$ is the Fréchet Lie group of smooth loops in the Lie group $G$. This thesis uses the caloron correspondence to construct certain differential f…

This work is based on the approach developed by J.~Dorfmeister, F.~Pedit and H.~Wu [GANG and KITCS preprint, Report KITCS94-4-1] to construct maps $Φ:D\rightarrow R^3$, $D$ being the unit disk in $C$, whose images are surfaces of constant mean curvature. They start from certain meromorphic one forms, so called meromorp…

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

This work extends elasticity theory to curved spaces, solving stress potentials.

Study properties of specific solitons on submanifolds with special vector fields.

The article derives some novel independence measures and contrast functions for Blind Source Separation (BSS) application. For the $k^{th}$ order differentiable multivariate functions with equal hyper-volumes (region bounded by hyper-surfaces) and with a constraint of bounded support for $k>1$, it proves that equality …

We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the…