English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show how Lasry-Lions's result on regularization of functions defined on$\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares ofdistances can be extended to (finite or infinite dimensional) Riemannianmanifolds $M$ of bounded sectional curvature. More specifically, among otherthings we show that if the sectional curvature $K$ of $M$ satisfies $-K_0\leqK\leq K_0$ on $M$ for some $K_0>0$, and if the injectivity and convexity radiiof $M$ are strictly positive, then every bounded, uniformly continuous function$f:M\to\mathbb{R}$ can be uniformly approximated by globally $C^{1,1}$functions defined by $$ (f_{\lambda})^{\mu}=\sup_{z\in M}\inf_{y\inM}\{f(y)+\frac{1}{2\lambda} d(z,y)^{2}-\frac{1}{2\mu}d(x,z)^2\} $$ as $\lambda,\mu\to 0^{+}$, with $0<\mu<\lambda/2$. Our definition of (global) $C^{1,1}$smoothness is intrinsic and natural, and it reduces to the usual one in flatspaces, but we warn the reader that, in the noncompact case, this definitiondiffers from other notions of (rather local) $C^{1,1}$ smoothness that havebeen recently used, for instance, by A. Fathi and P. Bernard (based on charts).The importance of this regularization method lies (rather than on the degree ofsmoothness obtained) on the fact that the correspondence $f\mapsto(f_{\lambda})^{\mu}$ is explicit and preserves many significant geometricalproperties that the given functions $f$ may have, such as invariance by a setof isometries, infima, sets of minimizers, ordering, local or globalLipschitzness, and (only when one additionally assumes that $K\leq 0$) local orglobal convexity. We also give two examples showing that this result completelyfails, even for (nonflat) Cartan-Hadamard manifolds, whenever $f$ or $K$ arenot bounded.

Regularization by sup-inf convolutions on Riemannian manifolds: an extension of Lasry-Lions theorem to manifolds of bounded curvature