Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Graphs with parallel mean curvature


Author: Isabel Maria da Costa Salavessa
Journal: Proc. Amer. Math. Soc. 107 (1989), 449-458
MSC: Primary 53C40; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1989-0965247-X
MathSciNet review: 965247
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if the graph $ {\Gamma _f} = \left\{ {\left( {x,f\left( x \right)} \right):x \in M} \right\}$ of a map $ f:\left( {M,g} \right) \to \left( {N,h} \right)$ between Riemannian manifolds is a submanifold of $ \left( {M \times N,g \times h} \right)$ with parallel mean curvature $ H$, then on a compact domain $ D \subset M$, $ \left\Vert H \right\Vert$ is bounded from above by $ \frac{1}{m}\frac{{A\left( {\partial D} \right)}}{{V\left( D \right)}}$. In particular, $ {\Gamma _f}$ is minimal provided $ M$ is compact, or noncompact with zero Cheeger constant. Moreover, if $ M$ is the $ m$-hyperbolic space--thus with nonzero Cheeger constant--then there exist real-valued functions the graphs of which are nonminimal submanifolds of $ M \times \mathbb{R}$ with parallel mean curvature.


References [Enhancements On Off] (What's this?)

  • [1] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, Florida, 1980. MR 768584 (86g:58140)
  • [2] S. S. Chern, On the curvatures of a piece of hypersurface in Euclidean space, Abh. Math. Sem. Univ. Hamburg 29 (1965), 77-91. MR 0188949 (32:6376)
  • [3] J. Eells, Minimal graphs, Manuscripta Math. 28 (1979), 101-108. MR 535698 (80i:58023)
  • [4] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 495450 (82b:58033)
  • [5] H. Flanders, Remark on mean curvature, J. London Math. Soc. (2) 41 (1966), 364-366. MR 0193600 (33:1818)
  • [6] M. Gaffney, A special Stokes' theorem for complete Riemannian manifolds, Ann. of Math. (2) 60 (1954), 141-145. MR 0062490 (15:986d)
  • [7] E. Heinz, Über Flächen mit Eineindeutigen Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen Eingeschränkt Sind, Math. Ann. 129 (1955), 451-454. MR 0071822 (17:189a)
  • [8] I. M. C. Salavessa, Graphs with parallel mean curvature and a variational problem in conformal geometry, Ph.D. Thesis, University of Warwick, 1988.
  • [9] S. T. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 451-454. MR 0417452 (54:5502)
  • [10] -, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. Paris (4) 8 (1987), 487-507. MR 0397619 (53:1478)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C40, 53C42

Retrieve articles in all journals with MSC: 53C40, 53C42


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0965247-X
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society