Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds
Tóm tắt
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru (Splitting of 3-manifolds and rigidity of area-minimizing surfaces,
arXiv:1107.5346
, 2011). We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.
Tài liệu tham khảo
Bray, H.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Thesis, Stanford University (1997)
Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. 18(4), 821–830 (2010)
Cai, M., Galloway, G.: Rigidity of area-minimizing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom. 8(3), 565–573 (2000)
Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. arXiv:1209.1165
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math. 33(2), 199–211 (1980)
Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996)
Kazdan, J., Warner, F.: Prescribing curvatures. In: Differential Geometry. Proc. Sympos. Pure Math., vol. 27, pp. 309–319. Am. Math. Soc., Providence (1975)
Ladyzhenskaia, O., Uralt’seva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968), 495 pp.
Li, M.: Rigidity of area-minimizing disks in three-manifolds with boundary. Preprint
Meeks, W., Yau, S.T.: Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. Math. (2) 112(3), 441–484 (1980)
Meeks, W., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179(2), 151–168 (1982)
Micallef, M., Moraru, V.: Splitting of 3-Manifolds and rigidity of area-minimizing surfaces. To appear in Proc. Am. Math. Soc. arXiv:1107.5346
Nunes, I.: Rigidity of area-minimizing hyperbolic surfaces in three-manifolds. J. Geom. Anal. (2011) doi:10.1007/s12220-011-9287-8. Published electronically
Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature. Ann. Math. (2) 110(1), 127–142 (1979)
Shen, Y., Zhu, S.: Rigidity of stable minimal hypersurfaces. Math. Ann. 309(1), 107–116 (1997)
Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra (1983), vii+272 pp.