Related papers: 1-Dimensional Intrinsic Persistence of Geodesic Sp…
The response of low-dimensional solid objects combines geometry and physics in unusual ways, exemplified in structures of great utility such as a thin-walled tube that is ubiquitous in nature and technology. Here we provide a particularly…
The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the…
We study stability properties of the topological entropy of Reeb flows on contact 3-manifolds with respect to the C^0-distance on the space of contact forms. Our main results show that a C^\infty-generic contact form on a closed co-oriented…
Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of…
In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the…
Persistent homology is a tool of topological data analysis that has been used in a variety of settings to characterize different dimensional holes in data. However, persistent homology computations can be memory intensive with a…
We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings on $C$ with the topology of pointwise convergence. We introduce the notion of a somewhat…
In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every…
Suppose that the surfaces K0 and Kr are the boundaries of two convex, complete, connected C^2 bodies in R^3. Assume further that the (Euclidean) distance between any point x in Kr and K0 is always r (r > 0). For x in Kr, let {\Pi}(x) denote…
We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form…
We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold…
This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…
Given any triplet of positive integers $n \geq 2$, $m$ and $k$ such that $n=m+k$, we exhibit a $C^1$ robustly transitive endomorphism of $\mathbb{T}^n$ with persistent critical points in the isotopy class of $F \times Id$, where $F$ is an…
Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric…
We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its…
This is an investigation into a classification of embeddings of a surface in Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi:\mathbb{R}^2 \times…
In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…
We investigate the geometry and topology of compact submanifolds of arbitrary codimension in space forms satisfying a certain pinching condition involving the length of the second fundamental form and the mean curvature. We prove that this…
We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…