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A local classification of spacelike surfaces in Minkowski 4-space, which are invariant under spacelike rotations, and with mean curvature vector either vanishing or lightlike, is obtained. Furthermore, the existence of such surfaces with…
An interface description and numerical simulations of model A kinetics are used for the first time to investigate the intra-surface kinetics of phase ordering on corrugated surfaces. Geometrical dynamical equations are derived for the…
We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed…
The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.
In this paper, we study the relation of the sign of the Gaussian and mean curvature of modular surfaces in Lorentz-Minkowski $3$-space to the zeroes of the associated complex analytic functions and its derivatives. Further, we completely…
Surfaces with concentric $K$-contours and parallel $K$-contours in Euclidean $3$-space are defined. Crucial examples are presented and characterization of them are given.
Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.
We give an immersion formula, the Sym-Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal…
Basic quantities related to 2-D gravity, such as Polyakov extrinsic action, Nambu-Goto action, geometrical action, and Euler characteristic are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces. Connection of the GWE…
Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.
Moduli spaces of stably irreducible sheaves on Kodaira surfaces belong to the short list of examples of smooth and compact holomorphic symplectic manifolds, and it is not yet known how they fit into the classification of holomorphic…
This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular…
For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of…
This paper studies deformations and birational maps between singular moduli spaces of semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that under certain conditions, two such moduli spaces of the same dimension…
Variational principles are developed within the framework of a spinor representation of the surface geometry to examine the equilibrium properties of a membrane or interface. This is a far-reaching generalization of the Weierstrass-Enneper…
We give a bound on which singularities may appear on Koll\'ar--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces…
We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations,…
We provide an intrinsic description of $N$-super \RS s and $TN$-\SR\ surfaces. Semirigid surfaces occur naturally in the description of topological gravity as well as topological supergravity. We show that such surfaces are obtained by an…
In this paper, by using a special Euler-Ramanujan identity and the idea of Wick rotation, we show that a one-parameter family of solutions to the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb E_1^3$, namely…
We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…