Inverse mean curvature flow over non-star-shaped surfaces

Harvie, Brian

doi:10.4310/mrl.2022.v29.n4.a7

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…Next, note that the inverse mean curvature flow (IMCF) for hypersurfaces yields a fully nonlinear parabolic equation with grow-up behaviour for a broad class of initial data. In particular, the evolution of star shaped (or more general) initial data under the rescaled ICMF converges to a round sphere, yielding certain stability result for the round sphere, see [22, 23]. In our approach, such a stability result can be interpreted as follows: the compactification of the star shaped (or more general grow-up) subset of phase-space of IMCF should have a unique stable equilibrium at infinity, which corresponds to the round sphere.…”

Section: Discussionmentioning

confidence: 99%

Unbounded Sturm attractors for quasilinear parabolic equations

Lappicy,

Fernandes

2024

Proceedings of the Edinburgh Mathematical Society

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We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

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Section: Discussionmentioning

confidence: 99%

Unbounded Sturm attractors for quasilinear parabolic equations

Lappicy,

Fernandes

2024

Proceedings of the Edinburgh Mathematical Society

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Isoperimetry and the properness of weak inverse mean curvature flow

2024

Calc. Var.

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The limit of the inverse mean curvature flow on a torus

Harvie

2022

Proc. Amer. Math. Soc.

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For an H > 0 H>0 rotationally symmetric embedded torus N 0 ⊂ R 3 N_{0} \subset \mathbb {R}^{3} evolved by Inverse Mean Curvature Flow, we show that the total curvature | A | |A| remains bounded up to the singular time T max T_{\max } . This in turn implies convergence of the N t N_{t} to a C 1 C^{1} rotationally symmetric embedded torus N T max N_{T_{\max }} as t → T max t \rightarrow T_{\max } without rescaling, contrasting sharply with the behavior of other extrinsic flows. Later, we note a scale-invariant L 2 L^{2} energy estimate on any flow solution in R 3 \mathbb {R}^{3} that may be useful in ruling out curvature blowup near singularities more generally.

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Inverse mean curvature flow over non-star-shaped surfaces

Cited by 3 publications

References 7 publications

Unbounded Sturm attractors for quasilinear parabolic equations

Unbounded Sturm attractors for quasilinear parabolic equations

Isoperimetry and the properness of weak inverse mean curvature flow

The limit of the inverse mean curvature flow on a torus

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