Relating diameter and mean curvature for Riemannian submanifolds
Authors:
JiaYong Wu and Yu Zheng
Journal:
Proc. Amer. Math. Soc. 139 (2011), 40974104
MSC (2010):
Primary 53C40, 57R42
Published electronically:
March 25, 2011
MathSciNet review:
2823054
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Abstract: Given an dimensional closed connected Riemannian manifold smoothly isometrically immersed in an dimensional Riemannian manifold , we estimate the diameter of in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of P. M. Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539546).
 1.
Bennett
Chow, Peng
Lu, and Lei
Ni, Hamilton’s Ricci flow, Graduate Studies in
Mathematics, vol. 77, American Mathematical Society, Providence, RI;
Science Press, New York, 2006. MR 2274812
(2008a:53068)
 2.
David
Hoffman and Joel
Spruck, Sobolev and isoperimetric inequalities for Riemannian
submanifolds, Comm. Pure Appl. Math. 27 (1974),
715–727. MR 0365424
(51 #1676)
David
Hoffman and Joel
Spruck, A correction to: “Sobolev and isoperimetric
inequalities for Riemannian submanifolds” (Comm. Pure Appl. Math. 27
(1974), 715–725), Comm. Pure Appl. Math. 28
(1975), no. 6, 765–766. MR 0397625
(53 #1484)
 3.
Haozhao
Li, The volumepreserving mean curvature flow in Euclidean
space, Pacific J. Math. 243 (2009), no. 2,
331–355. MR 2552262
(2010k:53100), 10.2140/pjm.2009.243.331
 4.
J.
H. Michael and L.
M. Simon, Sobolev and meanvalue inequalities on generalized
submanifolds of 𝑅ⁿ, Comm. Pure Appl. Math.
26 (1973), 361–379. MR 0344978
(49 #9717)
 5.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159.
 6.
Leon
Simon, Existence of surfaces minimizing the Willmore
functional, Comm. Anal. Geom. 1 (1993), no. 2,
281–326. MR 1243525
(94k:58028)
 7.
Peter
Topping, The optimal constant in Wente’s 𝐿^{∞}
estimate, Comment. Math. Helv. 72 (1997), no. 2,
316–328. MR 1470094
(98k:35075), 10.1007/s000140050018
 8.
Peter
Topping, Mean curvature flow and geometric inequalities, J.
Reine Angew. Math. 503 (1998), 47–61. MR 1650335
(99m:53080), 10.1515/crll.1998.099
 9.
Peter
Topping, Diameter control under Ricci flow, Comm. Anal. Geom.
13 (2005), no. 5, 1039–1055. MR 2216151
(2006m:53101)
 10.
Peter
Topping, Lectures on the Ricci flow, London Mathematical
Society Lecture Note Series, vol. 325, Cambridge University Press,
Cambridge, 2006. MR 2265040
(2007h:53105)
 11.
Peter
Topping, Relating diameter and mean curvature for submanifolds of
Euclidean space, Comment. Math. Helv. 83 (2008),
no. 3, 539–546. MR 2410779
(2009b:53100), 10.4171/CMH/135
 1.
 B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press, New York, 2006. MR 2274812 (2008a:53068)
 2.
 D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure. Appl. Math., 27 (1974), 715727. Erratum, Comm. Pure. Appl. Math., 28 (1975), 765766. MR 0365424 (51:1676); MR 0397625 (53:1484)
 3.
 H.Z. Li, The volumepreserving mean curvature flow in Euclidean space, Pacific J. Math., 243 (2009), 331355. MR 2552262 (2010k:53100)
 4.
 J. H. Michael and L. M. Simon, Sobolev and meanvalue inequalities on generalized submanifolds of , Comm. Pure Appl. Math., 26 (1973), 361379. MR 0344978 (49:9717)
 5.
 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159.
 6.
 L. M. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281326. MR 1243525 (94k:58028)
 7.
 P. M. Topping, The optimal constant in Wente's estimate, Comment. Math. Helv., 72 (1997), 316328. MR 1470094 (98k:35075)
 8.
 P. M. Topping, Mean curvature flow and geometric inequalities, J. Reine Angew. Math., 503 (1998), 4761. MR 1650335 (99m:53080)
 9.
 P. M. Topping, Diameter control under Ricci flow, Comm. Anal. Geom., 13 (2005), 10391055. MR 2216151 (2006m:53101)
 10.
 P. M. Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, 2006. MR 2265040 (2007h:53105)
 11.
 P. M. Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv., 83 (2008), 539546. MR 2410779 (2009b:53100)
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Additional Information
JiaYong Wu
Affiliation:
Department of Mathematics, Shanghai Maritime University, Haigang Avenue 1550, Shanghai 201306, People’s Republic of China
Email:
jywu81@yahoo.com
Yu Zheng
Affiliation:
Department of Mathematics, East China Normal University, Dong Chuan Road 500, Shanghai 200241, People’s Republic of China
Email:
zhyu@math.ecnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S000299392011108487
Keywords:
Mean curvature,
Riemannian submanifolds,
geometric inequalities,
diameter estimate.
Received by editor(s):
January 24, 2010
Received by editor(s) in revised form:
September 23, 2010
Published electronically:
March 25, 2011
Additional Notes:
This work is partially supported by NSFC10871069.
Communicated by:
Michael Wolf
Article copyright:
© Copyright 2011
American Mathematical Society
