Adiabatic theory for the area-constrained Willmore flow
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Adiabatic theory for the area-constrained Willmore flow. / Zhang, Jingxuan.
In: Journal of Mathematical Physics, Vol. 63, No. 4, 041503, 2022.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Adiabatic theory for the area-constrained Willmore flow
AU - Zhang, Jingxuan
PY - 2022
Y1 - 2022
N2 - In this paper, we develop an adiabatic theory for the evolution of large closed surfaces under the area-constrained Willmore (ACW) flow in a three-dimensional asymptotically Schwarzschild manifold. We explicitly construct a map, defined on a certain four-dimensional manifold of barycenters, which characterizes key static and dynamical properties of the ACW flow. In particular, using this map, we find an explicit four-dimensional effective dynamics of barycenters, which serves as a uniform asymptotic approximation for the (infinite-dimensional) ACW flow, so long as the initial surface satisfies certain mild geometric constraints (which determine the validity interval). Conversely, given any prescribed flow of barycenters evolving according to this effective dynamics, we construct a family of surfaces evolving by the ACW flow, whose barycenters are uniformly close to the prescribed ones on a large time interval (whose size depends on the geometric constraints of initial configurations).I. INTRODUCTION
AB - In this paper, we develop an adiabatic theory for the evolution of large closed surfaces under the area-constrained Willmore (ACW) flow in a three-dimensional asymptotically Schwarzschild manifold. We explicitly construct a map, defined on a certain four-dimensional manifold of barycenters, which characterizes key static and dynamical properties of the ACW flow. In particular, using this map, we find an explicit four-dimensional effective dynamics of barycenters, which serves as a uniform asymptotic approximation for the (infinite-dimensional) ACW flow, so long as the initial surface satisfies certain mild geometric constraints (which determine the validity interval). Conversely, given any prescribed flow of barycenters evolving according to this effective dynamics, we construct a family of surfaces evolving by the ACW flow, whose barycenters are uniformly close to the prescribed ones on a large time interval (whose size depends on the geometric constraints of initial configurations).I. INTRODUCTION
UR - https://doi.org/10.1063/5.0076701
U2 - 10.1063/5.0076701
DO - 10.1063/5.0076701
M3 - Journal article
VL - 63
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 4
M1 - 041503
ER -
ID: 304180219