In this paper we study singularities of certain surfaces and curves associated with the family of rectifying planes along space curves. We establish the relationships between singularities of these subjects and geometric invariants of curves which are deeply related to the order of contact with helices.

[For the entire collection see Zbl 0742.00067.]Let $P^m$ be the set of hyperplanes $\sigma :\langle \overrightarrow{x},\overrightarrow{\theta }\rangle =p$ in ${ℛ}^m$, $S^{m-1}$ the unit sphere of ${ℛ}^m$, $E^m$ the exterior of the unit ball, $T^m$ the set of hyperplanes not passing through the unit ball, $Rf(\overrightarrow{\theta },p)={\int }_{\sigma }f\left(\overrightarrow{x}\right)d\overrightarrow{x}$ the Radon transform, $R^{\#}g\left(\overrightarrow{x}\right)={\int }_{S^{m-1}}g(\overrightarrow{\theta },\langle \overrightarrow{x},\overrightarrow{\theta }\rangle )d{S}_{\overrightarrow{\theta }}$ its dual. $R$ as operator from $L^2\left({ℛ}^m\right)$ to $L^2(S^{m-1)}\times ℛ)$ is a closable, densely defined operator, $R^{*}$ denotes the operator given by $\left(R^{*}g\right)\left(\overrightarrow{x}\right)=R^{\#}g\left(\overrightarrow{x}\right)$ if the integral exists for $\overrightarrow{x}\in {ℛ}^m$ a.e. Then the closure of $R^{*}$ is the adjoint of $R$. The author shows that the Radon transform and its dual can be linked by two operators...

Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which may be either of short range or “gravitational” long range type. Thus, $g$ gives $X$ the geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be an operator of the form $H=\Delta +P$, where $\Delta$ is the Laplacian with respect to $g$ and $P$ is a self-adjoint first order scattering differential operator with coefficients vanishing at $\partial X$ and satisfying a “gravitational” condition. We define a symbol calculus for...

We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than $-1$. Without symmetry assumption, it holds if and only if the parameter is in the interval $(-1,0]$. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically...

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space $H_2^p(M,T^{*}M\otimes T^{*}M)$ with $p\>\frac{n}{2}$ and there exist a point $P\in M$ and $\delta \>0$ such that $g$ is smooth in the ball $B_p\left(\delta \right)$. We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to $g$ and of volume $1$. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...