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Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Critical Points of the Weighted Area Functional

Author : Qing-Ming Cheng

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

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Abstract

In this survey, we discuss critical points of functionals by various aspects. We review properties of critical points of weighted area functional, that is, self-shrinkers of mean curvature flow in Euclidean spaces and examples of compact self-shrinkers are discussed. We also review properties of critical points for weighted area functional for weighted volume-preserving variations, that is, \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean spaces.

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previous chapter Applications of the Affine Structures on the Teichmüller Spaces
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Literature
1.
Abresch, U., Langer, J.: The normalized curve shortening flow and homothetic solutions. J. Diff. Geom. 23, 175–196 (1986)MathSciNetMATH
2.
Angenent, S.: Shrinking doughnuts. In: Nonlinear diffusion equations and their equilibrium states, Birkhaüser, Boston-Basel-Berlin, vol. 7, pp. 21–38 (1992)
3.
4.
Calabi, E.: On manifolds with non-negative Ricci-curvature II. Notices Amer. Math. Soc. 22, A205 (1975)
5.
Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85, 175–185 (2010)MathSciNetMATH
6.
Cheng, Q.-M., Wei, G.: Complete \(\lambda \)-hypersurfaces of the weighted volume—preserving mean curvature flow, arXiv:​1403.​3177
7.
Cheng, Q.-M., Wei, G.: The weighted volume—preserving mean curvature flow, preprint
8.
9.
Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I. Generic Singularities. Ann. of Math. 175, 755–833 (2012)MathSciNetMATH
10.
11.
Drugan, G.: An immersed \(S^2\) self-shrinker, to appear in Trans. Amer. Math. Soc
12.
Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differential Geom. 26, 285–314 (1987)MathSciNetMATH
13.
14.
15.
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom. 22, 237–266 (1984)MathSciNetMATH
16.
17.
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Diff. Geom. 31, 285–299 (1990)MathSciNetMATH
18.
Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proceedings of Symposia in Pure Mathematics, Part 1, American Mathematical Society, Providence, RI, vol. 54, pp. 175–191(1993)
19.
Hamilton, R.S., Isoperimetric estimates for the curve shrinking flow in the plane. In: Modern Methods in Complex Analysis (Princeton, NJ, : Annals of Mathematics Studdies, Princeton University Press, Princeton, NJ 1995, vol. 137, pp. 201–222 (1992)
20.
Kapouleas, N., Kleene, S.J., Møller, N.M.: Mean curvature self-shrinkers of high genus: non-compact examples, to appear in J. Reine Angew. Math
21.
Kleene, S., Møller, N. M.: Self-shrinkers with a rotation symmetry, to appear in Trans. Amer. Math. Soc
22.
Li, H., Wei, Y.: Lower volume growth estimates for self-shrinkers of mean curvature flow. Proc. Amer. Math. Soc. 142, 3237–3248 (2014)MathSciNetCrossRefMATH
23.
24.
Munteanu, O., Wang, J.: Analysis of the weighted Laplacian and applications to Ricci solitons. Comm. Anal. Geom. 20, 55–94 (2012)MathSciNetCrossRefMATH
25.
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
Metadata
Title
Critical Points of the Weighted Area Functional
Author
Qing-Ming Cheng
Copyright Year
2016
Publisher
Springer Japan
Book
Geometry and Topology of Manifolds

Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-4-431-56021-0_4

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