Abstract
The partitioning problem for a smooth convex bodyB ⊂ ℝ3 consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.
We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.
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References
Anné, C.: Bornes sur la multiplicité, Preprint, 1992.
Aronszajn, N.: A unique continuation theorem for solution of elliptic partial differential equations or inequalities of second order,J. Math. Pures Appl. 36 (1957), 235–249.
Athanassenas, M.: A variational problem for constant mean curvature surfaces with free boundary,J. reine angew. Math. 377 (1987), 97–107.
Barbosa, J. L. and do Carmo, M.: Stability of hypersurfaces with constant mean curvature,Math. Z. 185 (1984), 339–353.
Barbosa, J. L., do Carmo, M. and Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds,Math. Z. 197 (1988), 123–138.
Bokowski, J. and Sperner Jr, E.: Zerlegung konvexer Körper durch minimale Trennflächen,J. reine angew. Math. 311/312 (1979), 80–100.
Cheng, S. Y.: Eigenfunctions and nodal sets,Comment. Math. Helv. 51 (1976), 43–55.
do Carmo, M.: Hypersurfaces of constant mean curvature,Proc. 3rd Symp. Diff. Geom., Peniscola, Lecture Notes in Math. 1410, Springer-Verlag, New York, 1988, pp. 128–144.
Dierkes, U., Hildebrandt, S., Küster, A. and Wohlrab, O.:Minimal Surfaces I, Springer-Verlag, Berlin, 1992.
El Soufi, A. and Ilias, S.: Majoration de la seconde valeur propre d'un operateur de Schrödinger sur une variété compacte et applications,J. Funct. Anal. 103 (1992), 294–316.
Griffiths, P. and Harris, J.:Principles of Algebraic Geometry, Wiley, New York, 1978.
Gilbart, D. and Trudinger, N.S.:Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
Grüter, M., Hildebrandt, S. and Nitsche, J. C. C.: Regularity for stationary surfaces of constant mean curvature with free boundaries,Acta Math. 156 (1986), 119–152.
Grüter, M. and Jost, J.: On embedded minimal disks in convex bodies,Ann. Inst. H. Poincaré. Anal. Non Linéaire 3 (1986), 345–390.
Grüter, M. and Jost, J.: Allard-type regularity results for varifolds with free boundaries,Ann. Sci. Norm. Sup. Pisa Cl. Sci., Ser. 4,13(1) (1986), 129–169.
Jost, J.: Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere,J. Differential Geom. 30 (1989), 555–577.
Li, P. and Yau, S. T.: A new conformal invariant and its applications to the Wilmore conjecture and the first eigenvalue of compact surfaces,Invent. Math. 69 (1982), 269–291.
Nitsche, J. C. C.: Stationary partitioning of convex bodies,Arch. Rational Mech. Anal. 89 (1985), 1–19.
Oliveira, G. de and Soret, M.: Personal communication.
Ritoré, M. and Ros, A.: Stable constant mean curvature tori and the isoperimetric problem in three space forms,Comment. Math. Helv. 67 (1992), 293–305.
Struwe, M.: The existence of surfaces of constant mean curvature with free boundaries,Acta Math. 160 (1988), 19–64.
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Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.