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.

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

Rotational λ – hypersurfaces in Euclidean spaces

2021 ◽  
Vol 30 (1) ◽  
pp. 29-40
Author(s):  
KADRI ARSLAN ◽  
ALIM SUTVEREN ◽  
BETUL BULCA
Keyword(s):  
Present Article ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Special Solution ◽  
Euclidean Spaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Flows

Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtained some results related with rotational hypersurfaces in Euclidean 4-space \mathbb{R}^{4} to become self-shrinkers. Furthermore, we classify the general rotational \lambda -hypersurfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational \lambda -hypersurfaces in \mathbb{R}^{4}.

scholarly journals A note on singular time of mean curvature flow

2009 ◽  
Vol 266 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Jingyi Chen ◽  
Weiyong He
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Singular Time

The Kähler-Ricci mean curvature flow of a strictly area decreasing map Between Riemann Surfaces

2020 ◽  
Vol 31 (08) ◽  
pp. 2050061
Author(s):  
Shujing Pan
Keyword(s):  
Scalar Curvature ◽  
Ricci Flow ◽  
Mean Curvature ◽  
Riemann Surfaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Positive Scalar Curvature ◽  
Curvature Decay ◽  
The Mean ◽  
Compact Riemann Surfaces

Suppose that [Formula: see text] is a product of compact Riemann surfaces [Formula: see text],[Formula: see text], i.e. [Formula: see text], and [Formula: see text] is a graph in [Formula: see text] of a strictly area dereasing map [Formula: see text]. Let [Formula: see text] evolve along the Kähler–Ricci flow, and [Formula: see text] in [Formula: see text] evolve along the mean curvature flow. We show that [Formula: see text] remains to be a graph of a strictly area decreasing map along the Kähler–Ricci mean curvature flow and exists for all time. In the positive scalar curvature case, we prove the convergence of the flow and the curvature decay along the flow at infinity.

scholarly journals A mean-curvature flow along a Kähler–Ricci flow

2018 ◽  
Vol 29 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Xiaoli Han ◽  
Jiayu Li ◽  
Liang Zhao
Keyword(s):  
Evolution Equation ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Initial Surface ◽  
Kähler Angle ◽  
Kähler Surface ◽  
The Mean

Let [Formula: see text] be a Kähler surface, and [Formula: see text] an immersed surface in [Formula: see text]. The Kähler angle of [Formula: see text] in [Formula: see text] is introduced by Chern and Wolfson [Am. J. Math. 105 (1983) 59–83]. Let [Formula: see text] evolve along the Kähler–Ricci flow, and [Formula: see text] in [Formula: see text] evolve along the mean-curvature flow. We show that the Kähler angle [Formula: see text] satisfies the evolution equation [Formula: see text] where [Formula: see text] is the scalar curvature of [Formula: see text]. The equation implies that if the initial surface is symplectic (Lagrangian), then, along the flow, [Formula: see text] is always symplectic (Lagrangian) at each time [Formula: see text], which we call a symplectic (Lagrangian) Kähler–Ricci mean-curvature flow. In this paper, we mainly study the symplectic Kähler–Ricci mean-curvature flow.

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