Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStable immersions in orbifolds
86
0
0.0
(
0
)
Authors
Alden Walker
Ask ChatGPT about the research
No Arabic abstract
We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari-Louwsma and resolve a conjecture of Calegari.
rate research
Read More
An immersion of a smooth $n$-dimensional manifold $M to mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M to mathbb{R}^{TN(M)}$. In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric bilinear maps. Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied. Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $mathbb{R}^{4n-1}$, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value $TN(mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. This is the first optimal-dimension result for any closed manifold $M eq S^1$, for any of the recently-studied configuration space conditions.
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.
We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.
We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichm{u}ller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on $3$-manifolds.
We show that Mirzakhanis curve counting theorem also holds if we replace surfaces by orbifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
