Energy identities and monotonicity for evolving 
                $$\varvec{k}$$
                
                    
                        
                            k
                        
                    
                
            -forms on moving Riemannian spaces

Afuni, Ahmad

doi:10.1007/s00028-017-0410-6

J. Evol. Equ.

2017

DOI: 10.1007/s00028-017-0410-6

|View full text |Cite

Energy identities and monotonicity for evolving $$\varvec{k}$$ k -forms on moving Riemannian spaces

Ahmad Afuni

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

2022

Publication Types

Select...

Article4

Relationship

Self Cite1

Independent3

Authors

Journals

Cited by 4 publications

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

Afuni

2021

J Geom Anal

Self Cite

View full text Add to dashboard Cite

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

show abstract

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

Afuni

2021

J Geom Anal

Self Cite

View full text Add to dashboard Cite

show abstract

Isoperimetric inequalities in unbounded convex bodies

Leonardi¹,

Ritoré²,

Vernadakis³

2022

Memoirs of the AMS

View full text Add to dashboard Cite

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body C ⊂ R n C\subset \mathbb {R}^{n} , without assuming any further regularity on the boundary of C C . Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of unbounded convex body with uniform geometry. We then provide a handy characterization of the uniform geometry property and, by exploiting the notion of asymptotic cylinder of C C , we prove existence of isoperimetric regions in a generalized sense. By an approximation argument we show the strict concavity of the isoperimetric profile and, consequently, the connectedness of generalized isoperimetric regions. We also focus on the cases of small as well as of large volumes; in particular we show existence of isoperimetric regions with sufficiently large volumes, for special classes of unbounded convex bodies. We finally address some questions about isoperimetric rigidity and analyze the asymptotic behavior of the isoperimetric profile in connection with the notion of isoperimetric dimension.

show abstract

Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

Zheng

2021

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We show that the isoperimetric profile h g ( t ) ( ξ ) h_{g(t)}(\xi ) of a compact Riemannian manifold ( M , g ) (M,g) is jointly continuous when metrics g ( t ) g(t) vary continuously. We also show that, when M M is a compact surface and g ( t ) g(t) evolves under normalized Ricci flow, h g ( t ) 2 ( ξ ) h^2_{g(t)}(\xi ) is uniform Lipschitz continuous and hence h g ( t ) ( ξ ) h_{g(t)}(\xi ) is uniform locally Lipschitz continuous.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Energy identities and monotonicity for evolving $$\varvec{k}$$ k -forms on moving Riemannian spaces

Cited by 4 publications

References 16 publications

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

Isoperimetric inequalities in unbounded convex bodies

Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

Contact Info

Product

Resources

About