On 1-Harmonic Functions

Shihshu Walter Wei

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over $\mathbb{R}$; and every 7-dimensional $SO(2)\times SO(6)$-invariant absolutely area-minimizing integral current in $\mathbb{R}^8$ is real analytic. The assumption on the $SO(2)\times SO(6)$-invariance cannot be removed, due to the first counter-example in $\mathbb{R}^8$, proved by Bombieri, De Girogi and Giusti.

Détail

Keywords: 1-harmonic function; 1-tension field; absolutely area-minimizing integral current.

Bibliographie

[1] Andreotti A., Vesentini E., “Carleman estimate for the Laplace–Beltrami equation on complex manifolds”, Inst. Hautes Études Sci. Publ. Math., 25 (1965), 81–130 | DOI | MR

[2] Brothers J. E., “Invariance of solutions to invariant parametric variational problems”, Trans. Amer. Math. Soc., 262 (1980), 159–179 | DOI | MR | Zbl

[3] Bombieri E., de Giorgi E., Giusti E., “Minimal cones and the Bernstein problem”, Invent. Math., 7 (1969), 243–268 | DOI | MR | Zbl

[4] Federer H., “Some theorems on integral currents”, Trans. Amer. Math. Soc., 117 (1965), 43–67 | DOI | MR | Zbl

[5] Federer H., Geometric measure theory, Springer, Berlin – Heidelberg – New York, 1969 | MR

[6] Federer H., “The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension”, Bull. Amer. Math. Soc., 76 (1970), 767–771 | DOI | MR | Zbl

[7] Federer H., “Real flat chains, cochains and variational problems”, Indiana Univ. Math. J., 24 (1974), 351–406 | DOI | MR

[8] Federer H., Fleming W. H., “Normal and integral currents”, Ann. of Math. (2), 72 (1960), 458–520 | DOI | MR | Zbl

[9] Hsiang W. Y., “On the compact homogeneous minimal submanifolds”, Proc. Natl. Acad. Sci. USA, 56 (1966), 5–6 | DOI | MR | Zbl

[10] Hsiang W. Y., Lawson H. B. Jr., “Minimal submanifolds of low cohomogeneity”, J. Differential Geom., 5 (1971), 1–38 | MR | Zbl

[11] Karp L., “Subharmonic functions on real and complex manifolds”, Math. Z., 179 (1982), 535–554 | DOI | MR | Zbl

[12] Lawson H. B., “The stable homology of a flat torus”, Math. Scand., 36 (1975), 49–73 | MR | Zbl

[13] Lawson H. B., “The equivariant Plateau problem and interior regularity”, Trans. Amer. Math. Soc., 173 (1972), 231–249 | DOI | MR | Zbl

[14] Lin F.-H., “Minimality and stability of minimal hypersurfaces in $R^N$”, Bull. Austral. Math. Soc., 36 (1987), 209–214 | DOI | MR | Zbl

[15] Miranda M., “Sul minimo dell'integrale del gradiente di una funzione”, Ann. Scuola Norm. Sup. Pisa (3), 19 (1965), 626–665 | MR

[16] Miranda M., “Comportamento delle successioni convergenti di frontiere minimali”, Rend. Sem. Mat. Univ. Padova, 39 (1967), 238–257 | MR

[17] Simoes P., A class minimal cones in $\mathbb R^n,n\ge8$, that minimizes area, Thesis, Berkeley, 1973 | MR

[18] Simons J., “Minimal varieties in riemannian manifolds”, Ann. of Math. (2), 88 (1968), 62–105 | DOI | MR | Zbl

[19] Wang S. P., Wei S. W., “Bernstein conjecture in hyperbolic geometry”, Seminar on Minimal Submanifolds, Ann. of Math. Stud., 103, ed. E. Bombieri, Princeton Univ. Press, Princeton, 1983, 339–358 | MR

[20] Wei S. W., Minimality, stability, and Plateau's problem, Thesis, Berkeley, 1980

[21] Wei S. W., “Plateau's problem in symmetric spaces”, Nonlinear Anal., 12 (1988), 749–760 | DOI | MR | Zbl

[22] Yau S. T., “Some function theoretic properties of complete Riemannian manifolds and their applications to geometry”, Indiana Univ. Math. J., 25 (1976), 659–670 | DOI | MR | Zbl