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Geometric properties of the space of Lagrangian self-shrinking tori in ℝ⁴ Ma, Man Shun
Abstract
We prove that any sequence {Fn : ∑ → ℝ⁴} of conformally branched compact Lagrangian self-shrinkers to the mean curvature flow with uniform area upper bound has a convergent subsequence, if the conformal structures do not degenerate. When ∑ has genus one, we can drop the assumption on non-degeneracy the conformal structures. When ∑ has genus zero, we show that there is no branched immersion of ∑ as a Lagrangian self-shrinker, generalizing the rigidity result of [52] in dimension two by allowing branch points. When the area bound is small, we show that any such Lagrangian self-shrinking torus in $\mathbb R^4$ is embedded with uniform curvature estimates. For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori, along which the Lagrangian condition is preserved, area is decreasing, and the compact type I singularities with a fixed area upper bound can be perturbed away in finitely many steps. This is a Lagrangian version of the construction for embedded surfaces in ℝ³ in [17]. In the noncompact situation, we derive a parabolic Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on ℓ-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental form. This generalizes a result of Wang [53] for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in [8].
Item Metadata
Title |
Geometric properties of the space of Lagrangian self-shrinking tori in ℝ⁴
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2017
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Description |
We prove that any sequence {Fn : ∑ → ℝ⁴} of conformally branched compact Lagrangian self-shrinkers to the mean curvature flow with uniform area upper bound has a convergent subsequence, if the conformal structures do not degenerate. When ∑ has genus one, we can drop the assumption on non-degeneracy the conformal structures. When ∑ has genus zero, we show that there is no branched immersion of ∑ as a Lagrangian self-shrinker, generalizing the rigidity result of [52] in dimension two by allowing branch points.
When the area bound is small, we show that any such Lagrangian self-shrinking torus in $\mathbb R^4$ is embedded with uniform curvature estimates.
For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Łojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori.
Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori, along which the Lagrangian condition is preserved, area is decreasing, and the compact type I singularities with a fixed area upper bound can be perturbed away in finitely many steps. This is a Lagrangian version of the construction for embedded surfaces in ℝ³ in [17].
In the noncompact situation, we derive a parabolic Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on ℓ-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental form. This generalizes a result of Wang [53] for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in [8].
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Genre | |
Type | |
Language |
eng
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Date Available |
2017-05-23
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0347551
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2017-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International