Willmore Functionals

Author:
Mingliang Cai

Journal:
Proc. Amer. Math. Soc. **127** (1999), 569-575

MSC (1991):
Primary 53C20

DOI:
https://doi.org/10.1090/S0002-9939-99-04484-6

MathSciNet review:
1458864

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove some integral inequalities for immersed tori in the three sphere. The functionals considered are generalizations of the Willmore functional.

**[C]**Chen, B.Y.*Geometry of submanifolds,*Marcel Dekker, New York, 1973. MR**50:5697****[CG]**Cai, M. and Galloway, G.*Least area tori and manifolds of nonnegative scalar curvature,*Math. Z., 223(1996), pp.387-395. MR**97i:53040****[H]**Hutchinson, J.*Second fundamental form for varifolds and the existence of surfaces minimizing curvature,*Indiana U. Math. J., 35(1986), pp.45-71. MR**87e:48068****[Lan]**Langer, J.*A compactness theorem for surfaces with bounded second fundamental form,*Math. Ann., 270(1985), pp.223-234. MR**86i:53035****[Law]**Lawson, B.*Local rigidity theorems for minimal surfaces,*Ann. of Math., 89(1969), pp.187-197.**[LS]**Langer, J. and Singer, D.*Curves in the hyperbolic plane and mean curvatures of tori in 3 space,*Bull. London Math. Soc., 16(1984), pp.531-534. MR**85k:53006****[LY]**Li, P. and Yau, S.T.*A new conformal invariant and its applications to the Willmore conjecture and first eigenvalues of compact surfaces,*Invent. Math., 69(1982), pp.269-291. MR**84f:53049****[MR]**Montiel, S. and Ros, A.*Minimal immersions of surfaces by the first eigenfunctions and conformal area,*Invent. Math., 83(1985), pp.153-166. MR**87d:53109****[P]**Pinkall, U.*Hopf tori in ,*Invent. Math., 81(1985), pp. 379-386. MR**86k:53075****[S]**Simon, L.*Existence of surfaces minimizing the Willmore functional,*Comm. Anal. Geom. 1(1993), pp.281-326. MR**94k:58028**

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Additional Information

**Mingliang Cai**

Affiliation:
Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124

Email:
mcai@math.miami.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-04484-6

Keywords:
Principal curvature,
exponential map

Received by editor(s):
January 30, 1997

Received by editor(s) in revised form:
May 19, 1997

Communicated by:
Christopher Croke

Article copyright:
© Copyright 1999
American Mathematical Society