L2 harmonic forms and stability of minimal hypersurfaces

Shukichi TANNO

doi:10.2969/jmsj/04840761

L² harmonic forms and stability of minimal hypersurfaces

Shukichi TANNO

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 4 Pages 761-768

DOI https://doi.org/10.2969/jmsj/04840761

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Published: 1996 Received: February 28, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) F. G. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein theorem, Ann. of Math., 84 (1966), 277-292.
2) S. Bernstein, Sur un théorème de Géomérie et ses applications aux équations aux dérivées partielles du type elliptique, Comm. Soc. Math. Kharkov, 5 (1915-17), 38-45.
3) E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-269.
4) Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren Math. Wiss., 285, Springer-Verlag, 1988.
5) E. De Giorgi, Una extensione del theoreme di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85.
6) M, do Carmo and C. K. Peng, Stable minimal surfaces in R³ are planes, Bull. Amer. Math. Soc., 1 (1979), 903-906.
7) J. Dodziuk, L² harmonic forms on complete manifolds, Ann, of Math. Stud., 102 (1982), 291-302.
8) D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math., 33 (1980), 199-211.
9) R. Miyaoka, Harmonic 1-forms on a complete stable minimal hypersurface, (preprint).
10) B. Palmer, Stability of minimal hypersurfaces, Comm. Math. Helv., 66 (1991), 185-188.
11) R. Schoen and S. T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, Ann. of Math. Stud., 102 (1982), 209-228.
12) J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
13) S. Tanno, The topology of contact Riemannian manifolds, Illinois J. Math., 12 (1968), 700-717.
14) S. Tanno, Instability of spheres with deformed Riemannian metrics, Kodai Math. J., 10 (1987), 250-257.

Right : [1] F. G. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein theorem, Ann. of Math., 84 (1966), 277-292.
[2] S. Bernstein, Sur un théorème de Géomérie et ses applications aux équations aux dérivées partielles du type elliptique, Comm. Soc. Math. Kharkov, 5 (1915-17), 38-45.
[3] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7 (1969), 243-269.
[4] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren Math. Wiss., 285, Springer-Verlag, 1988.
[5] E. De Giorgi, Una extensione del theoreme di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85.
[6] M, do Carmo and C. K. Peng, Stable minimal surfaces in R³ are planes, Bull. Amer. Math. Soc., 1 (1979), 903-906.
[7] J. Dodziuk, L² harmonic forms on complete manifolds, Ann. of Math. Stud., 102 (1982), 291-302.
[8] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math., 33 (1980), 199-211.
[9] R. Miyaoka, Harmonic 1-forms on a complete stable minimal hypersurface, (preprint).
[10] B. Palmer, Stability of minimal hypersurfaces, Comm. Math. Helv., 66 (1991), 185-188.
[11] R. Schoen and S. T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, Ann. of Math. Stud., 102 (1982), 209-228.
[12] J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.
[13] S. Tanno, The topology of contact Riemannian manifolds, Illinois J. Math., 12 (1968), 700-717.
[14] S. Tanno, Instability of spheres with deformed Riemannian metrics, Kodai Math. J., 10 (1987), 250-257.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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