scholarly journals Evolution of convex entire graphs by curvature flows

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Roberta Alessandroni ◽  
Carlo Sinestrari
Keyword(s):  
Euclidean Space ◽  
Initial Data ◽  
Mean Curvature ◽  
Symmetric Function ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Principal Curvatures ◽  
The Mean ◽  
Entire Graphs

AbstractWe consider the evolution of an entire convex graph in euclidean space with speed given by a symmetric function of the principal curvatures. Under suitable assumptions on the speed and on the initial data, we prove that the solution exists for all times and it remains a graph. In addition, after appropriate rescaling, it converges to a homothetically expanding solution of the flow. In this way, we extend to a class of nonlinear speeds the well known results of Ecker and Huisken for the mean curvature flow.

NUMERICAL EVIDENCE OF FATTENING FOR THE MEAN CURVATURE FLOW

1996 ◽  
Vol 06 (06) ◽  
pp. 793-813 ◽  
Author(s):  
FRANCESCA FIERRO ◽  
MAURIZIO PAOLINI
Keyword(s):  
Numerical Simulations ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Normal Direction ◽  
Compact Surface ◽  
Principal Curvatures ◽  
Forcing Term ◽  
The Mean ◽  
Motion By Mean Curvature

In this paper we describe some numerical simulations in the context of mean curvature flow. We recover a few different approaches in modeling the evolution of an interface Σ which evolves according to the law: V=κ+g where V is the velocity in the inward normal direction, κ is the sum of the principal curvatures and g is a given forcing term. We will discuss about the phenomenon of fattening or nonuniqueness of the solution, recalling what is known about this subject. Finally we show some interesting numerical simulations that suggest evidence of fattening starting from different initial interfaces. Of particular interest is the result obtained for a torus in ℝ4 which would be a first example of a regular and compact surface showing evidence of fattening in the case of pure motion by mean curvature (no forcing term).

scholarly journals Rigidity results and topology at infinity of translating solitons of the mean curvature flow

2017 ◽  
Vol 19 (06) ◽  
pp. 1750002 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Minimal Hypersurfaces ◽  
Rigidity Results ◽  
And Topology ◽  
The Mean ◽  
Translating Solitons

In this paper, we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of [Formula: see text]-minimal hypersurfaces.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals Finite topology self-translating surfaces for the mean curvature flow in R3

2017 ◽  
Vol 320 ◽  
pp. 674-729 ◽  
Author(s):  
Juan Dávila ◽  
Manuel del Pino ◽  
Xuan Hien Nguyen
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Finite Topology ◽  
The Mean

The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ 2 {\mathbb{C}^{2}}

2018 ◽  
Vol 2018 (743) ◽  
pp. 229-244 ◽  
Author(s):  
Jingyi Chen ◽  
John Man Shun Ma
Keyword(s):  
Upper Bound ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Riemannian Metric ◽  
Branch Points ◽  
Rigidity Result ◽  
The Mean ◽  
Uniform Area

Abstract Let F_{n} : (Σ, h_{n} ) \to \mathbb{C}^{2} be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics \{ h_{n} \} converges smoothly to a Riemannian metric h. We show that a subsequence of \{ F_{n} \} converges smoothly to a branched conformally immersed Lagrangian self-shrinker F_{\infty} : (Σ, h) \to \mathbb{C}^{2} . When the area bound is less than 16π, the limit {F_{\infty}} is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence h_{n} \to h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.

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}.

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