scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

scholarly journals Ancient solutions of the Ricci flow on bundles

2017 ◽  
Vol 318 ◽  
pp. 411-456 ◽  
Author(s):  
Peng Lu ◽  
Y.K. Wang
Keyword(s):  
Ricci Flow ◽  
Ancient Solutions

Collapsing ancient solutions of mean curvature flow

2021 ◽  
Vol 119 (2) ◽  
Author(s):  
Theodora Bourni ◽  
Mat Langford ◽  
Giuseppe Tinaglia
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Ancient Solutions

scholarly journals Optimal Curvature Estimates for Homogeneous Ricci Flows

2017 ◽  
Vol 2019 (14) ◽  
pp. 4431-4468 ◽  
Author(s):  
Christoph Böhm ◽  
Ramiro Lafuente ◽  
Miles Simon
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Convergence Result ◽  
Einstein Metric ◽  
Type I ◽  
Extinction Time ◽  
Ricci Flows ◽  
Gap Theorem ◽  
Ancient Solutions ◽  
Curvature Estimates

AbstractWe prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n)({\mathrm{scal}}(g(t)) - {\mathrm{scal}}(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, with constants depending only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on that space. The above curvature estimates follow from a gap theorem for Ricci-flatness on homogeneous spaces. This theorem is proved by contradiction, using a local $W^{2,p}$ convergence result which holds without symmetry assumptions.

scholarly journals Ancient solutions of superlinear heat equations on Riemannian manifolds

2020 ◽  
pp. 2050033
Author(s):  
Daniele Castorina ◽  
Carlo Mantegazza
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Heat Equations ◽  
Qualitative Properties ◽  
Ancient Solutions

We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.

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