We classify simply-connected, orientable, complete Willmore surfaces with vanishing Gaussian curvature. also study the cones in $${\mathbb {R}}^{3}$$ and give a classification. As an application, we show that for embedding $$f:{\mathbb {R}}^{2} \rightarrow {\mathbb , if its corresponding curvature is nonnegative image of Gauss map lies $$\big \{\theta \in {S}}^{2}: d_{{\mathbb {S}}^{2}}(\theta ...

Contents 1. Introduction 1 1.1. Notation and Conventions 2 1.2. Brill-Noether varieties and Abel maps 4 1.3. Stability and semistability 6 2. Technical groundwork 9 2.1. Basic estimates 9 2.2. Basic cases 12 2.3. Divisors imposing independent conditions 14 3. Irreducibility and dimension 19 3.1. Irreducible components of the Theta divisor 19 3.2. Dimension of the image of the Abel map 24 4. Com...

for any lacunary sequence $theta = (k_{r})$, we define the concepts of $s_{theta}-$limit point and $s_{theta}-$cluster point of a sequence of fuzzy numbers $x = (x_{k})$. we introduce the new sets  $lambda^{f}_{s_{theta}}(x)$, $gamma^{f}_{s_{theta}}(x)$ and prove some inclusion relaions between these and the sets $lambda^{f}_{s}(x)$, $gamma^{f}_{s}(x)$ introduced in ~cite{ayt:slpsfn} by aytar [...

More than forty years ago, Igusa proved in [Ig] a “Fundamental Lemma”, according to which, theta constants give an explicit generically injective map from some modular varieties related to the symplectic group Sp(n, R) into the projective space. The embedded varieties satisfy some quartic relations, the so called Riemann’s relations. We expect that similar results hold for orthogonal instead of...

X linked retinoschisis (RS) causes poor vision in affected males owing to radial cystic changes at the macula. Genetic linkage analysis was carried out in 16 British families with X linked retinoschisis using markers from the Xp22 region. Linkage was confirmed between the RS locus and the markers DXS207 (lod score, Zmax = 17.9 at recombination fraction theta = 0.03; confidence interval for thet...

In this paper, we study the period map of a certain one-parameter family of quartic K3 surfaces with an S5-action. We construct automorphic forms on the period domain as the pull-backs of theta constants of genus 2 by a modular embedding. Using these automorphic forms, we give an explicit presentation of the inverse period map.

For a harmonic map $u : M^3 \to S^1$ on closed, oriented $3$-manifold, we establish the identity \[ 2 \pi \int_{\theta \in S^1} \chi (\Sigma _\theta) \geq \frac{1}{2} \int_{\Sigma_\theta} \left( {\lvert du \rvert}^{-2} \mathit{Hess} (u) \rvert}^2 + R_M \right) \] relating scalar curvature $R_M$ of $M$ to average Euler characteristic level sets $\Sigma_\theta = u^{-1} {\lbrace \theta \rbrace}$. ...