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Universidad de Granada
 

Lectures on stable minimal surfaces

Granada (España)

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In the first part of these lectures, the aim is the topological and geometric classification of complete stable $H-$surfaces immersed in a manifold $(\amb , g)$ whose Scalar curvature is nonnegative. Here, we will show the H. Rosenberg's diameter estimate [Rosen06] (following ideas of D. Fischer-Colbrie [Fis85]) and the classification of complete stable minimal surfaces given by D. Fischer- Colbrie and R. Schoen [FS80] and R. Schoen and S.T. Yau [SY82]. We take the point of view of stable Schrodinger operators as in the work of Meeks-Pérez-Ros [MPR08].

In the second part, we classify manifolds $(\amb , g)$ under the existence of certain compact area minimizing surface and a lower bound of its Scalar curvature. We will show area estimates for stable compact minimal surfaces and in the case that estimated is attained, we will show how the manifold splits locally around such an area minimizing surface. In the case we also add conditions saying that such a surface is area minimizing on its homotopy class and attains the estimate, the splitting is global. The idea is to extend the splitting theorems developed by Cai-Galloway [CG00], Bray-Brendle-Neves [BBN10], I. Nunes [Nun12]. Here, we will take the unified point of view considered by Micallef-Moraru [MM].

Conferencia financiado por el programa de doctorado Matemáticas (MHE2011-00248).

References

[BBN10]
H. Bray, S. Brendle, A. Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom., 18 (2010), 821-830.
[CG00]
M. Cai and G. Galloway, Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature, Comm. Anal. Geom., 8 (2000), 565-573.
[Esp10]
J. M. Espinar, Rigidity of stable cylinders in three-manifolds. To appear in Proc. A.M.S.
[Fis85]
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. Math., 82 (1985), 121-132.
[FS80]
D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure and Appl. Math., 33 (1980), 199-211.
[MPR08]
W. Meeks, J. Pérez and A. Ros, Stable constant mean curvature hypersurfaces, Handbook of Geometric Analisys, volume 1 (2008), pages 381-380. International Press, edited by Lizhen Ji, Peter Li, Richard Schoen and Leon Simon, ISBN: 978-1-57146-130-8.
[MM]
M. Micallef, V. Moraru, Splitting of three-manifolds and rigidity of area-minimizing surfaces. Preprint.
[Nun12]
I. Nunes, Rigidity of Area-Minimizing hyperbolic surfaces in three-manifolds. To appear in J. Geom. Anal.
[Rosen06]
H.Rosenberg, Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Austral. Math. Soc.,74 (2006), 227-238.
[SY82]
R. Schoen, S.T. Yau, Complete three dimensional manifolds with positive Ricci curvature and scalar curvature, Ann. Math. Stud., 102 (1982), 209-228.

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