Related papers: Willmore tori in the 4-sphere with nontrivial norm…
In this paper we prove a relative version of the classical Mumford-Newstead theorem for a family of smooth curves degenerating to a reducible curve with a simple node. We also prove a Torelli-type theorem by showing that certain moduli…
We give an example of an eversion of the 2-sphere in the Euclidean 3-space, inspired by Morse theory, with a unique quadruple point. No homotopical tool is used.
This paper is a continuation of a paper with the same title of the last two authors. In the first part of the present paper, we give a unified geometric proof that both focal submanifolds of every isoparametric hypersurface in spheres with…
The classification of Willmore 2-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n=3,4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n=5$. There…
We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber…
We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the…
Non-trivial examples of Riemannian almost product structures are constructed on the product bundle of the positive and negative twistor spaces of an oriented Riemannian four-manifold. The Gil-Medrano and Naveira types of these structures…
Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.
In this paper we show that locally there exists a Willmore deformation between minimal surfaces in $S^{n+2}$ and minimal surfaces in $H^{n+2}$, i.e., there exists a smooth family of Willmore surfaces $\{y_t,t\in[0,1]\}$ such that…
We study tori attached to the fundamental groups of plane curves with arbitrary singularities. These tori provide complete information about homology of finite abelian covers of the plane branched along the curve. We calculate these tori in…
We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.
The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $\nu$-gonal curve"), concerning classification…
This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…
We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal…
It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and…
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…
Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is…
The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of…
Let $ X: M \hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \W(X)$, and study its critical points called Willmore Legendrian…
The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of…