Short-time existence of the α-Dirac-harmonic map flow and applications

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

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Moving mesh for the axisymmetric harmonic map flow

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an:2005034 ◽

2005 ◽

Vol 39 (4) ◽

pp. 781-796

Author(s):

Benoit Merlet ◽

Morgan Pierre

Keyword(s):

Harmonic Map ◽

Moving Mesh ◽

Harmonic Map Flow

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Short time existence for the elastic flow of clamped curves

Mathematische Nachrichten ◽

10.1002/mana.201700368 ◽

2018 ◽

Vol 291 (13) ◽

pp. 2115-2116

Keyword(s):

Short Time ◽

Time Existence

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Asymptotics of the Teichmüller harmonic map flow

Advances in Mathematics ◽

10.1016/j.aim.2013.05.021 ◽

2013 ◽

Vol 244 ◽

pp. 874-893 ◽

Cited By ~ 10

Author(s):

Melanie Rupflin ◽

Peter M. Topping ◽

Miaomiao Zhu

Keyword(s):

Harmonic Map ◽

Harmonic Map Flow

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Ricci flow on manifolds with boundary with arbitrary initial metric

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0060 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Tsz-Kiu Aaron Chow

Keyword(s):

Ricci Flow ◽

Positive Curvature ◽

Curvature Operator ◽

Manifolds With Boundary ◽

Convex Boundary ◽

Uniqueness Of The Solution ◽

Positive Time ◽

Natural Boundary Conditions ◽

Short Time ◽

Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

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