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.

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 prove that any cyclic quadrilateral can be inscribed in any closed convex $C^{1}$ -curve. The smoothness condition is not required if the quadrilateral is a rectangle.

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$ -neighborhood of the unit ball, the only solutions of $\unicode[STIX]{x1D6F1}^{2}K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$ -dimensional convex body, $\unicode[STIX]{x1D6F1}K$ is the projection body of $K$ and $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

We apply a mean-value inequality for positive subsolutions of the $f$ -heat operator, obtained from a Sobolev embedding, to prove a nonexistence result concerning complete noncompact $f$ -maximal spacelike hypersurfaces in a class of weighted Lorentzian manifolds. Furthermore, we establish a new Calabi–Bernstein result for complete noncompact maximal spacelike hypersurfaces in a Lorentzian product space.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries $\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$ . The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$ , for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$ . In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$ . Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$ , then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$ -module ${\mathcal{O}}(X)$ . We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$ -representation that is closely related to the $\mathfrak{g}$ -modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$ . Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$ .

In Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596], we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based on the Cotton–York tensor in dimension 3 and the Weyl tensor in dimension 4. In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension 3 or 4, we classify the possible Cotton–York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits an LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596] but not admitting LCW.

We introduce a new affinely invariant structure on smooth surfaces in ℝ3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ‘ points’, which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.

We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

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 obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.