Related papers: Distortion Minimal Morphing I: The Theory For Stre…

We define linear and radial stretch maps in the affine-additive group, and prove that they are minimizers of the mean quasiconformal distortion functional. For the proofs we use a method based on the notion of modulus of a curve family and…

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the…

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how…

We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable…

Let $M$ be a closed surface and $f$ a diffeomorphism of $M$. A diffeomorphism is said to permute a dense collection of domains, if the union of the domains are dense and the iterates of any one domain are mutually disjoint. In this note, we…

In [CKM17], Chodosh, Ketover, and Maximo proved finite diffeomorphism theorems for complete embedded minimal hypersurfaces of dimension $\leqslant$ 6 with finite index and bounded volume growth ratio. In this paper, we adapt their method to…

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by…

In this exploration paper, we design algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. Such a contraction is a kind of shrinking or reducing process. In our algorithms, we need to…

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization…

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are…

A Morse-Bott volume form on a manifold is a top-degree form which vanishes along a non-degenerate critical submanifold. We prove that two such forms are diffeomorphic (by a diffeomorphism fixed on the submanifold) provided that their…

In the process of projecting the surface of a three-dimensional object onto a two-dimensional surface, due to the perspective distortion, the image on the surface of the object will have different degrees of distortion according to the…

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation…

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between…

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms.…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest…

We consider defining the embedding of a triangle mesh into $R^3$, up to translation, rotation, and scale, by its vector of dihedral angles. Theoretically, we show that locally, almost everywhere, the map from realizable vectors of dihedrals…

In optical imaging, light propagation is affected by the inhomogeneities of the medium. Sample-induced aberrations and multiple scattering can strongly degrade the image resolution and contrast. Based on a dynamic correction of the incident…