Smooth Approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leq\epsilon(p)$ for every $p\in M$ and $\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.Comment: 10 page

The Morse-Sard theorem revisited

Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n, \mathbb{R}^m)$ functions with $p>n$ and, on the other hand, also the following new result: if $f\in C^{k-1}(\mathbb{R}^n, \mathbb{R}^m)$ satisfies $$\limsup_{h\to 0}\frac{|D^{k-1}f(x+h)-D^{k-1}f(x)|}{|h|}<\infty$$ for every $x\in\mathbb{R}^n$ (that is, $D^{k-1}f$ is a Stepanov function), then the set of critical values of $f$ is Lebesgue-null in $\mathbb{R}^m$. In the case that $m=1$ we also show that this limiting condition holding for every $x\in\mathbb{R}^n\setminus\mathcal{N}$, where $\mathcal{N}$ is a set of zero $(n-2+\alpha)$-dimensional Hausdorff measure for some $0<\alpha<1$, is sufficient to guarantee the same conclusion.Comment: We corrected some misprints and made some changes in the introductio

Can we make a Finsler metric complete by a trivial projective change?

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.Comment: 11 pages, one figure, submitted to the proceedings of VI International Meeting on Lorentzian Geometry (Granada

Combined homogeneous and heterogeneous hydrogenation to yield catalyst-free solutions of parahydrogen-hyperpolarized [1-13C]succinate

We show that catalyst-free aqueous solutions of hyperpolarized [1-13C]succinate can be produced using parahydrogen-induced polarization (PHIP) and a combination of homogeneous and heterogeneous catalytic hydrogenation reactions. We generate hyperpolarized [1-13C]fumarate via PHIP using para-enriched hydrogen gas with a homogeneous ruthenium catalyst, and subsequently remove the toxic catalyst and reaction side products via a purification procedure. Following this, we perform a second hydrogenation reaction using normal hydrogen gas to convert the fumarate into succinate using a solid Pd/Al2O3 catalyst. This inexpensive polarization protocol has a turnover time of a few minutes, and represents a major advance for in vivo applications of [1-13C]succinate as a hyperpolarized contrast agent

Smooth extensions of functions on separable Banach spaces

Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.Comment: 19 pages. This version fixes a gap in the previous proof of Theorem 1 by providing a sharp version of Lemma