A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds

Annibale Magni

Geometric Flows (2015)

  • Volume: 1, Issue: 1, page 609-639
  • ISSN: 2353-3382

Abstract

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We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.

How to cite

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Annibale Magni. " A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds ." Geometric Flows 1.1 (2015): 609-639. <http://eudml.org/doc/275889>.

@article{AnnibaleMagni2015,
abstract = {We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.},
author = {Annibale Magni},
journal = {Geometric Flows},
keywords = {Higher order geometric flows; Willmore functional; geometric measure theory; mean curvature flow; reduced Allen-Cahn action functional; action minimization problem; compactness; lower semicontinuity; Euler-Lagrange equation; symmetries},
language = {eng},
number = {1},
pages = {609-639},
title = { A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds },
url = {http://eudml.org/doc/275889},
volume = {1},
year = {2015},
}

TY - JOUR
AU - Annibale Magni
TI - A convergence result for the Gradient Flow of ∫ |A| 2 in Riemannian Manifolds
JO - Geometric Flows
PY - 2015
VL - 1
IS - 1
SP - 609
EP - 639
AB - We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.
LA - eng
KW - Higher order geometric flows; Willmore functional; geometric measure theory; mean curvature flow; reduced Allen-Cahn action functional; action minimization problem; compactness; lower semicontinuity; Euler-Lagrange equation; symmetries
UR - http://eudml.org/doc/275889
ER -

References

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  16. [16] A. Mondino and T. Riviére. Willmore Spheres in Compact Riemannian Manifolds. Adv. Math., 232(1):608–676, 2013. [WoS] Zbl1294.30046
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