Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds

2010 ◽  
Vol 41 (1-2) ◽  
pp. 111-125 ◽  
Author(s):  
Mykhaylo Chursin ◽  
Lars Schäfer ◽  
Knut Smoczyk
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Lagrangian Submanifolds ◽  
Curvature Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds

scholarly journals f-Minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

2019 ◽  
Author(s):  
Wei-Bo SU
Keyword(s):  
Vector Fields ◽  
Mean Curvature Flow ◽  
Lagrangian Submanifolds ◽  
Soliton Solutions ◽  
Curvature Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Variation Formula ◽  
Special Case ◽  
Translating Solitons

Abstract The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind including minimal Lagrangians and soliton solutions for Lagrangian mean curvature flow (LMCF). We derive 2nd variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh’s formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

scholarly journals Convergence of mean curvature flow in hyper-Kähler manifolds

2020 ◽  
Vol 305 (2) ◽  
pp. 667-691
Author(s):  
Keita Kunikawa ◽  
Ryosuke Takahashi
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds

scholarly journals Mean curvature flow of monotone Lagrangian submanifolds

2007 ◽  
Vol 257 (2) ◽  
pp. 295-327 ◽  
Author(s):  
K. Groh ◽  
M. Schwarz ◽  
K. Smoczyk ◽  
K. Zehmisch
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Lagrangian Submanifolds ◽  
Curvature Flow

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 Mean curvature flow with free boundary outside a hypersphere

2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow
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