We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha >2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak’s $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

Let $f\;:\; M\rightarrow \mathbb{C}P^{2}$ be an isometric immersion of a compact surface in the complex projective plane $\mathbb{C}P^{2}$. In this paper, we consider the Helfrich-type functional $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)=\int _{M}(|H|^{2}+\lambda _{1}+\lambda _{2} C^{2})\textrm{d} M$, where $\lambda _{1}, \lambda _{2}\in \mathbb{R}$ with $\lambda _{1}\geqslant 0$, $H$ and $C$ are respectively the mean curvature vector and the Kähler function of $M$ in $\mathbb{C}P^{2}$. The critical surfaces of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ are called Helfrich surfaces. We compute the first variation of $\mathcal{H}_{\lambda _{1},\lambda _{2}}(f)$ and classify the homogeneous Helfrich tori in $\mathbb{C}P^{2}$. Moreover, we study the Helfrich energy of the homogeneous tori and show the lower bound of the Helfrich energy for such tori.

We obtain a complete topological classification of $k$-folding map-germs on generic surfaces in $\mathbb {R}^3$, discover new robust features of surfaces and recover, in a unified way, many of the robust features that were obtained previously by considering the contact of a surface with lines, planes or spheres.

We study ribbons of vanishing Gaussian curvature, i.e. flat ribbons, constructed along a curve in $\mathbb {R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the difference between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.

Le nombre chromatique relatif $c_0(S)$ d’une surface compacte S à bord est défini comme la borne supérieure des nombres chromatiques des graphes plongés dans S avec tous leurs sommets sur $\partial S$ . Cet invariant topologique a été introduit pour l’étude de la multiplicité de la première valeur propre de Steklov sur S. Dans cet article, on montre que $c_0(S)$ est aussi pertinent pour l’étude des plongements polyédraux tendus de S en établissant deux résultats. Le premier est que s’il existe un plongement polyédral tendu de S dans $\mathbb {R}^n$ qui n’est pas contenu dans un hyperplan, alors $n\leq c_0(S)-1$ . Le second est que cette inégalité est optimale pour les surfaces de petit genre.

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 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 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 characterise singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ${{\mathbb{R}}^{3}}$ that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in ${{H}^{3}}$ or ${{S}^{3}}$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.