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Register a new userIsometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces
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Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.
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We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.
We consider a class of compact homogeneous CR manifolds, that we call $mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3tomathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.
Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.
We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $mathbb D$ in $mathbb C$ into the unit ball $mathbb B^n$ in $mathbb R^n$, $nge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $nge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $mathbb Dto mathbb B^n$. This implies that conformal harmonic immersions $M to mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$mathrm{e}$ metric on $M$ and the Cayley-Klein metric on the ball $mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $mathbb D$ onto affine discs in $mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $mathbb R^n$ based on minimal surfaces.
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m leq pleq infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riemannian manifold $M$ with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.
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