arXiv research

Constructs a moment map flow for isotropic maps on surfaces.

Study isotropic Riemannian maps and helices along them.

The study examines surfaces in isotropic space with specific Gauss map properties.

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by Hélein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valu…

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piece…

Study on stability of geodesic maps in non-isotropic manifolds.

In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is…

Study isotropic curves on complex quadric with geometric relations.

In [5], together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from $2$-spheres to the unitary group $U(n)$ in terms of freely chosen meromorphic functions on $S^2$. The simplest harmonic maps are the isotropic ones. Using Morse theory Burstall and Guest [1] showed that the harmo…

Survey of twistor lifts of surfaces in 4-dimensional spaces.

Study stabilizers of isotropic classes in rational 4-manifolds, finding diffeomorphisms that almost preserve Lefschetz fibrations.

The paper classifies Willmore 2-spheres in $S^n$.

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

We show that no exotic $\mathbb{R}^4$ admits a complete Riemannian metric with uniformly positive isotropic curvature and with bounded geometry. This is essentially a corollary of the main result in [Hu1], and was stated in [Hu2] without proof. In the process of the proof we also show that the diffeomorphism type of an…

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $ρ\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of $\mathbb{O}$ to $S^6$. This allows us to associate to each surface $Σ$ of $\mathbb{O}$ a fun…

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in $\mathbb R^{n+2}$, isotropic surfaces in $S^4$ and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $S^6$ without dual surfaces is …

This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite ty…

We solve the analogue of Björling's problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $S^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the c…

Extends a formula for the homomorphism defect of a signature map to coloured braids.

The paper examines Randers metrics with isotropic scalar curvature properties.

Paper establishes a relation between Berwald scalar curvature and S-curvature.

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (…

In this paper, we find a condition on $(α, β)$-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.

Study physical work done by isotropic vector forces along isotropic curves.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

Study proves rigidity of harmonic maps from 2-torus to complex projective space.

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…

Spinor representation in isotropic space via Laguerre geometry.

The paper studies hanging chains and surfaces in degenerate geometries.

Maps between positively curved manifolds with non-increasing area are rigid.

We create real-time geodesic rendering for non-isotropic geometries.

Let $\mathcal{G}$ be a parahoric group scheme over a complex projective curve $X$ of genus greater than one. Let $\mathrm{Bun}_{\mathcal{G}}$ denote the moduli stack of $\mathcal{G}$-torsors on $X$. We prove several results concerning the Hitchin map on $T^*\!\mathrm{Bun}_{\mathcal{G}}$. We first show that the parahori…

Developed a new concept of isometric surfaces in isotropic space.

Paper shows isotropic $E$- and $S$-curvatures are equivalent in warped Finsler metrics.

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isot…

The paper studies Kropina metrics with a specific curvature property.

We will investigate the local geometry of the surfaces in the $7$-dimensional Euclidean space associated to harmonic maps from a Riemann surface $Σ$ into $S^6$. By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions $\varphi:Σ\to S^6$ whose associated immersio…

New approach uses isotropic geometry to solve Euclidean problems.

We show that for $m>n\geq 2$, there are at least two exact isotropic $n$-tori in $\mathbb{C}^m$ which are not Hamiltonian isotopic in $\mathbb{C}^m$, even though they are smoothly isotopic as isotropic $n$-tori. We apply this discovery to obtain more distinct non-exact isotropic tori in $\mathbb{C}^m$.

Study classifies zero mean curvature surfaces with planar curvature lines.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

This work extends holomorphic surface representations to isotropic space.

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.

Paper classifies Randers metrics based on Ricci curvature properties.

The study finds compact vacuum static spaces with positive isotropic curvature are spheres or products of a circle and sphere.

Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $γ: R\to R^{n+1,n}$ is {\it isotropic} if $γ, γ_x, \ldots, γ_x^{(2n)}$ are linearly independent and the span of $γ, \ldots, γ_x^{(n-1)}$ is …

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a gen…