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In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.
We study equilibrium surfaces for an energy which is a linear combination of the area and a second term which measures the bending and twisting of the boundary. Specifically, the twisting energy is given by the twisting of the Darboux frame. This energy is a modification of the Euler–Plateau functional considered by Giomi and Mahadevan (2012, Proc. R. Soc. A 468, 1851–1864), and a natural special case of the Kirchhoff–Plateau energy considered by Biria and Fried (2014, Proc. R. Soc. A 470, 20140368; 2015, Int. J. Eng. Sci. 94, 86–102).
A Willmore surface
$y:M\rightarrow S^{n+2}$
has a natural harmonic oriented conformal Gauss map
$Gr_{y}:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$
, which maps each point
$p\in M$
to its oriented mean curvature 2-sphere at
$p$
. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition, which will be called “strongly conformally harmonic.” The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface
$M$
to
$SO^{+}(1,n+3)/SO^{+}(1,3)\times SO(n)$
, which are the conformal Gauss maps of some Willmore surface in
$S^{n+2}.$
It turns out that generically, the condition of being strongly conformally harmonic suffices to be associated with a Willmore surface. The exceptional case will also be discussed.
In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.
As was shown by a part of the authors, for a given
$(2,3,5)$
-distribution
$D$
on a five-dimensional manifold
$Y$
, there is, locally, a Lagrangian cone structure
$C$
on another five-dimensional manifold
$X$
which consists of abnormal or singular paths of
$(Y,D)$
. We give a characterization of the class of Lagrangian cone structures corresponding to
$(2,3,5)$
-distributions. Thus, we complete the duality between
$(2,3,5)$
-distributions and Lagrangian cone structures via pseudo-product structures of type
$G_{2}$
. A local example of nonflat perturbations of the global model of flat Lagrangian cone structure which corresponds to
$(2,3,5)$
-distributions is given.
Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in
$3$
-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in
$d$
-space can be cut into
$(r-1)(d+1)+1$
pieces that can be rearranged by translations to form
$r$
loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension
$d\geqslant 3$
with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order
$2$
or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in
$C^{2}$
by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.
We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.
A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.
In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold
$G/P$
. The key to this approach is that in each case
$G/P$
is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group
$G$
with fixed point group
$H$
gives rise to a large family of such compactifications of homogeneous spaces of
$H$
. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces
$\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$
and
$\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$
. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.
In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.
Image feature points are detected as pixels which locally maximise a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris–Stephens corner detector. A major limitation of these feature detectors is that they are only Euclidean-invariant. In this work, we demonstrate the application of a 2D equi-affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.
We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.
We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.
We study local properties of the Bakry–Émery curvature function
${\mathcal{K}}_{G,x}:(0,\infty ]\rightarrow \mathbb{R}$
at a vertex
$x$
of a graph
$G$
systematically. Here
${\mathcal{K}}_{G,x}({\mathcal{N}})$
is defined as the optimal curvature lower bound
${\mathcal{K}}$
in the Bakry–Émery curvature-dimension inequality
$CD({\mathcal{K}},{\mathcal{N}})$
that
$x$
satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and
$S^{1}$
-out regularity, and relate the curvature functions of
$G$
with various spectral properties of (weighted) graphs constructed from local structures of
$G$
. We prove that the curvature functions of the Cartesian product of two graphs
$G_{1},G_{2}$
are equal to an abstract product of curvature functions of
$G_{1},G_{2}$
. We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy
$CD(0,\infty )$
but are not Cayley graphs.
We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.
We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.
We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.
In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in
$\mathbb{R}^{n}$
and in minimally convex domains of
$\mathbb{R}^{n}$
, results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.