Related papers: Real forms of minimal SL$_2$-threefolds

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable…

We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum,…

We investigate the homotopy type of a certain homogeneous space for a simple complex algebraic group. We calculate some of its classical topological invariants and introduce a new one. We also propose several conjectures about its…

The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism…

In this paper, we give a complete description of the deformation classes of real structures on minimal ruled surfaces. In particular, we show that these classes are determined by the topology of the real structure, which means that real…

The quintic threefold $X$ is the most studied Calabi-Yau $3$-fold in the mathematics literature. In this paper, using \v{C}ech-to-derived spectral sequences, we investigate the mod $2$ and integral cohomology groups of a real Lagrangian…

This paper includes the classification, in a simple Lie algebra, of the singularities of Slodowy slices between special nilpotent orbits that are adjacent in the partial order on nilpotent orbits. The irreducible components of most…

Let $T$ be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules $V$ with the following property: for each point $v\in V$ the closure $\bar{Tv}$ of its…

Building on a classical solution to the pentagon equation, constructed earlier by the author and E.V. Martyushev and related to the flat geometry invariant under the group SL(2), we construct an algebraic complex corresponding to a…

We prove some fundamental structural results for spherical varieties in arbitrary characteristic. In particular, we study Luna's two types of localization and use it to analyze spherical roots, colors and their interrelation. At the end, we…

We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure…

By analogy with associative and co-associative cases we introduce a class of three-dimensional non-orientable submanifolds, of almost $\mathrm{G}_2-$manifolds, modelled on planes lying in a special $\mathrm{G}_2-$orbit. An application of…

We describe the real forms of Gizatullin surfaces of the form $xy=p(z)$ and of Koras-Russell threefolds of the first kind. The former admit zero, two, three, four or six isomorphism classes of real forms, depending on the degree and the…

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

We introduce a unified framework for counting representations of knot groups into $SU(2)$ and $SL(2, \mathbb{R})$. For a knot $K$ in the 3-sphere, Lin and others showed that a Casson-style count of $SU(2)$ representations with fixed…

This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under…

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal…

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total…

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…