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Volume 37, Issue 1
Gradient Flow of the $L_β$-Functional

Xiaoli Han, Jiayu Li & Jun Sun

DOI: 10.4208/cmr.2020-0037

Commun. Math. Res., 37 (2021), pp. 113-140.

Published online: 2021-02

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  • Abstract

In this paper, we start to study the gradient flow of the functional $L_β$ introduced by Han-Li-Sun in [8]. As a first step, we show that if the initial surface is symplectic in a Kähler surface, then the symplectic property is preserved along the gradient flow. Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form. When $β$=1, we derive a monotonicity formula for the flow. As applications, we show that the $λ$-tangent cone of the flow consists of the finite flat planes.

  • Keywords

$β$-symplectic critical surfaces, gradient flow, monotonicity formula, tangent cone.

  • AMS Subject Headings

53C42, 53C44

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hanxiaoli@tsinghua.edu.cn (Xiaoli Han)

jiayuli@ ustc.edu.cn (Jiayu Li)

sunjun@whu.edu.cn (Jun Sun)

  • BibTex
  • RIS
  • TXT
@Article{CMR-37-113, author = {Han , XiaoliLi , Jiayu and Sun , Jun}, title = {Gradient Flow of the $L_β$-Functional}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {37}, number = {1}, pages = {113--140}, abstract = {

In this paper, we start to study the gradient flow of the functional $L_β$ introduced by Han-Li-Sun in [8]. As a first step, we show that if the initial surface is symplectic in a Kähler surface, then the symplectic property is preserved along the gradient flow. Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form. When $β$=1, we derive a monotonicity formula for the flow. As applications, we show that the $λ$-tangent cone of the flow consists of the finite flat planes.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0037}, url = {http://global-sci.org/intro/article_detail/cmr/18627.html} }
TY - JOUR T1 - Gradient Flow of the $L_β$-Functional AU - Han , Xiaoli AU - Li , Jiayu AU - Sun , Jun JO - Communications in Mathematical Research VL - 1 SP - 113 EP - 140 PY - 2021 DA - 2021/02 SN - 37 DO - http://doi.org/10.4208/cmr.2020-0037 UR - https://global-sci.org/intro/article_detail/cmr/18627.html KW - $β$-symplectic critical surfaces, gradient flow, monotonicity formula, tangent cone. AB -

In this paper, we start to study the gradient flow of the functional $L_β$ introduced by Han-Li-Sun in [8]. As a first step, we show that if the initial surface is symplectic in a Kähler surface, then the symplectic property is preserved along the gradient flow. Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form. When $β$=1, we derive a monotonicity formula for the flow. As applications, we show that the $λ$-tangent cone of the flow consists of the finite flat planes.

Han , XiaoliLi , Jiayu and Sun , Jun. (2021). Gradient Flow of the $L_β$-Functional. Communications in Mathematical Research . 37 (1). 113-140. doi:10.4208/cmr.2020-0037
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