Area‐minimizing projective planes in 3‐manifolds

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Rigidity of area-minimizing two-spheres in three-manifolds (2010, background): “with positive scalar curvature. By a theorem of Schoen and Yau [12], any area-minimizing surface in M is homeomorphic to either S2 or RP2. The case of area-minimizing projective planes was studied in [3]. In particular, if Σ is an embedded RP2 of minimal area, then the area of Σ can be estimated fro”
Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds (2011, background): “eaminimizing in its homotopy class and has area equal to 4π, then the universal cover of M is isometric to (R×S2,dt2+g). A similar rigidity result for areaminimizing projective planes was obtained in [3]. A natural question is to know what happens when the model is the Riemannian product manifold (R”
Rigidity of min-max minimal spheres in three-manifolds (2011, methodology): “holds if and only if g has constant sectional curvature one. The proof of the rigidity statement uses short-time existence for Hamilton’s Ricci ﬂow and the Maximum Principle, in the same spirit as in [3]. Connections between Ricci ﬂow and the theory of minimal surfaces have been previously considere”
Minimal 2-spheres in 3-spheres (2017, background): “1) is achieved by an embedded minimal projective plane Σ0 (see for instance Proposition 5 in [4]).”