Smooth approximation of Lipschitz functions on Riemannian manifolds

Azagra, Daniel; Ferrera, Juan; López-Mesas, Fernando; Rangel, Yenny C.

doi:10.1016/j.jmaa.2006.03.088

Cited by 61 publications

(72 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Removing the factor c in the above estimate is one main purpose of this paper. In fact we obtain the following R n ω n ≥ 1 K 2n (2) where…”

Section: Statement Of Resultsmentioning

confidence: 98%

See 1 more Smart Citation

Volume growth of some uniformly contractible Riemannian manifolds

Liu

2006

Math. Z.

View full text Add to dashboard Cite

It is shown that if a uniformly contractible Riemannian n-manifoldwhere ω n is the volume of the unit Euclidean ball. In particular, if M is uniformly contractible and d GH ((M, d g ), (V n , · )) < ∞, then M has at least Euclidean volume growth. This corollary covers an earlier result by Burago and Ivanov. Our results are motivated by a volume growth theorem contained in Gromov's book [Gromov in Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999, p. 256], we give a detailed proof of this theorem. Using the same argument, we also derive a generalization of the theorem which is pointed out by Gromov.

show abstract

“…Removing the factor c in the above estimate is one main purpose of this paper. In fact we obtain the following R n ω n ≥ 1 K 2n (2) where…”

Section: Statement Of Resultsmentioning

confidence: 98%

“…The following approximation theorem is given in [15, Prop. 2.1, p. 59], also see [2,14]. Now we introduce the notion of systolic degree, which is due to Gromov [16,p.…”

Section: A Generalization Of Theorem 35mentioning

confidence: 99%

Volume growth of some uniformly contractible Riemannian manifolds

Liu

2006

Math. Z.

View full text Add to dashboard Cite

show abstract

“…The Lipschitz nullhomotopy can be approximated by a smooth Lipschitz nullhomotopy (e.g., ref. 4). By the proof of the Poincaré lemma, this shows that the volume form on R n is dα for some bounded form α.…”

Section: Optimistic Possibilitymentioning

confidence: 87%

Quantitative algebraic topology and Lipschitz homotopy

Ferry

Weinberger

2013

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

We consider when it is possible to bound the Lipschitz constant a priori in a homotopy between Lipschitz maps. If one wants uniform bounds, this is essentially a finiteness condition on homotopy. This contrasts strongly with the question of whether one can homotop the maps through Lipschitz maps. We also give an application to cobordism and discuss analogous isotopy questions.amenable group | uniformly finite cohomology T he classical paradigm of geometric topology, exemplified by, at least, immersion theory, cobordism, smoothing and triangulation, surgery, and embedding theory is that of reduction to algebraic topology (and perhaps some additional pure algebra). A geometric problem gives rise to a map between spaces, and solving the original problem is equivalent to finding a nullhomotopy or a lift of the map. Finally, this homotopical problem is solved typically by the completely nongeometric methods of algebraic topology (e.g., localization theory, rational homotopy theory, spectral sequences).Although this has had enormous successes in answering classical qualitative questions, it is extremely difficult [as has been emphasized by Gromov (1)] to understand the answers quantitatively. One general type of question that tests one's understanding of the solution of a problem goes like this: Introduce a notion of complexity, and then ask about the complexity of the solution to the problem in terms of the complexity of the original problem. Other possibilities can involve understanding typical behavior or the implications of making variations of the problem.In this task, the complexity of the problem is often reflected, somewhat imperfectly, in the Lipschitz constant of the map. Indeed, one can often view the Lipschitz constant of the map as a measure of the complexity of the geometric problem.For concreteness, let us quickly review the classical case of cobordism, following Thom (2). Let M be a compact smooth manifold. The problem is: When is M n the boundary of some other compact smooth W n+1 ? There are many possible choices of manifolds in this construction, such as oriented manifolds, manifolds with some structure on their (stabilized) tangent bundles, Piecewise linear (PL) and topological versions, and so on. However, for now, we will confine our attention to this simplest version.Thom (2) embeds M m in a high-dimensional Euclidean space M ⊂ S m+N−1 ⊂ R m+N and then classifies the normal bundle by a map νM: νM → E(ξ N ↓ Gr(N, m + N)) from the normal bundle to the universal bundle of N-planes in m + N space. Including R m+N into S m+N via one-point compactification, we can think of νM as being a neighborhood of M in this sphere (via the tubular neighborhood theorem) and extend this map to S m+N if we include E(ξ N ↓ Gr(N, m + N)) into its one-point compactification E(ξ N ↓ Gr(N, m + N))^, the Thom space of the universal bundle. Let us call this mapThom (2) , extending the embedding of M into S m+N . Extending Thom's construction over this disk gives a nullhomotopy of Φ M . Conversely, one uses the nullhomotop...

show abstract

“…Fortunately, Lipschitz functions with constant c can be approximated uniformly by functions that are infinitely differentiable (socalled C ∞ -functions) with the same Lipschitz constant c. One way to do this is through a technique known as "mollification"; see e.g. Azagra et al (2007Azagra et al ( , pp. 1370Azagra et al ( -1372 for details.…”

Section: Proof Of Theorem 3 Inmentioning

confidence: 99%